What mechanism at the microscopic level determines whether a system heats up or not?












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When placed in an ordinary or a microwave oven, a beaker of water heats up except during boiling (i.e., a phase change involving latent heat). Now, suppose a system absorbs energy in such a way that the electrons are excited to higher energy levels. Will this necessarily heat up the system? In other words, is heating caused by the electrons or atoms in an object being excited to higher energy levels?










share|cite|improve this question











$endgroup$

















    3












    $begingroup$


    When placed in an ordinary or a microwave oven, a beaker of water heats up except during boiling (i.e., a phase change involving latent heat). Now, suppose a system absorbs energy in such a way that the electrons are excited to higher energy levels. Will this necessarily heat up the system? In other words, is heating caused by the electrons or atoms in an object being excited to higher energy levels?










    share|cite|improve this question











    $endgroup$















      3












      3








      3


      1



      $begingroup$


      When placed in an ordinary or a microwave oven, a beaker of water heats up except during boiling (i.e., a phase change involving latent heat). Now, suppose a system absorbs energy in such a way that the electrons are excited to higher energy levels. Will this necessarily heat up the system? In other words, is heating caused by the electrons or atoms in an object being excited to higher energy levels?










      share|cite|improve this question











      $endgroup$




      When placed in an ordinary or a microwave oven, a beaker of water heats up except during boiling (i.e., a phase change involving latent heat). Now, suppose a system absorbs energy in such a way that the electrons are excited to higher energy levels. Will this necessarily heat up the system? In other words, is heating caused by the electrons or atoms in an object being excited to higher energy levels?







      quantum-mechanics thermodynamics statistical-mechanics solid-state-physics phase-transition






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 22 '18 at 20:12









      Geoffrey

      3,66911229




      3,66911229










      asked Dec 22 '18 at 15:03









      mithusengupta123mithusengupta123

      1,28111435




      1,28111435






















          3 Answers
          3






          active

          oldest

          votes


















          3












          $begingroup$

          The microwave photon energy level gives rotational kinetic energy to water molecules due to interaction of the electromagnetic field with the water dipole molecule. The rotational kinetic energy is subsequently randomized to increase the average translational kinetic energy of the molecules, increasing the temperature of the water molecules. The microwave energy levels are well below those needed to excite electrons to higher levels.



          A regular oven cooks with infrared electromagnetic energy. The photon energy level corresponds to molecular vibration. These energy levels are likewise too low for electron excitation.



          Hope this helps






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            A toaster oven certainly uses radiation, but I'd argue that a bake-a-cake oven also uses conduction and convection.
            $endgroup$
            – psitae
            Dec 22 '18 at 20:17






          • 2




            $begingroup$
            @psitae What's your point? Whether its conduction, convection, radiation (microwave or infrared) none of these cook by exciting electrons.
            $endgroup$
            – Bob D
            Dec 22 '18 at 21:26










          • $begingroup$
            Why mention photons in this context?
            $endgroup$
            – Pieter
            Dec 23 '18 at 19:11



















          3












          $begingroup$

          On the case of a microwave oven the water molecules in the food are made to vibrate. Water molecules are tiny electric dipoles and these are made to vibrate more strongly. The molecules stay in the electronic ground state.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            What happens in ordinary ovens/burners? For heating to take place, a necessary condition is that the system must absorb energy. For this to happen, there must be a coupling between the environment that supplies energy and the system that absorbs it. For the microwave example, it is the electromagnetic microwave that couples to the dipole moment of water molecules.
            $endgroup$
            – mithusengupta123
            Dec 22 '18 at 17:14










          • $begingroup$
            The microwave photon energy level causes rotation not vibration of water dipole molecules. The vibration energy is higher than the microwave oven photon energy. For a good explanation of the interaction of radiation with matter, check out the Hyperphysics web site
            $endgroup$
            – Bob D
            Dec 22 '18 at 17:24












          • $begingroup$
            @BobD "Photon" does not explain anything here. The electric field polarizes the dipole orientation which relaxes when the field is switched off again. The field changes direction $2.45cdot10^9$ times per second, so that becomes noticeable as heat.
            $endgroup$
            – Pieter
            Dec 23 '18 at 19:17












          • $begingroup$
            @Pieter With all due respect, photon energy explains why microwave and infrared radiation do not excite electrons resulting in cooking, which I thought the OP was wondering. It also explains why the electric field in the kilohertz range does not cause molecule rotation.
            $endgroup$
            – Bob D
            Dec 24 '18 at 14:43





















          0












          $begingroup$

          The heat contained in a system can be thought of as the total kinetic energy of a system. my2cts' answer is correct, but I'll add that if you excite an electron to a higher energy level, that will indeed cause heating in many cases (see for example FRET).



          The friction between the water molecules and the rest of the system is what causes things to heat up in a microwave. In a conventional oven, it's a direct transfer of kinetic energy from hot air.



          Homework question: would something heat up in a microwave if it didn't contain any water?






          share|cite|improve this answer









          $endgroup$









          • 2




            $begingroup$
            Heat is energy transfer from one substance to another solely due to a temperature difference between them. Systems do not "contain" heat. You cannot think of heat as the total kinetic energy of a system. The proper term for the total kinetic energy (plus potential energy) is internal energy.
            $endgroup$
            – Bob D
            Dec 22 '18 at 21:31












          • $begingroup$
            @BobD that's correct. I'll edit my answer.
            $endgroup$
            – psitae
            Dec 22 '18 at 21:34











          Your Answer





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          3 Answers
          3






          active

          oldest

          votes








          3 Answers
          3






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          3












          $begingroup$

          The microwave photon energy level gives rotational kinetic energy to water molecules due to interaction of the electromagnetic field with the water dipole molecule. The rotational kinetic energy is subsequently randomized to increase the average translational kinetic energy of the molecules, increasing the temperature of the water molecules. The microwave energy levels are well below those needed to excite electrons to higher levels.



          A regular oven cooks with infrared electromagnetic energy. The photon energy level corresponds to molecular vibration. These energy levels are likewise too low for electron excitation.



          Hope this helps






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            A toaster oven certainly uses radiation, but I'd argue that a bake-a-cake oven also uses conduction and convection.
            $endgroup$
            – psitae
            Dec 22 '18 at 20:17






          • 2




            $begingroup$
            @psitae What's your point? Whether its conduction, convection, radiation (microwave or infrared) none of these cook by exciting electrons.
            $endgroup$
            – Bob D
            Dec 22 '18 at 21:26










          • $begingroup$
            Why mention photons in this context?
            $endgroup$
            – Pieter
            Dec 23 '18 at 19:11
















          3












          $begingroup$

          The microwave photon energy level gives rotational kinetic energy to water molecules due to interaction of the electromagnetic field with the water dipole molecule. The rotational kinetic energy is subsequently randomized to increase the average translational kinetic energy of the molecules, increasing the temperature of the water molecules. The microwave energy levels are well below those needed to excite electrons to higher levels.



          A regular oven cooks with infrared electromagnetic energy. The photon energy level corresponds to molecular vibration. These energy levels are likewise too low for electron excitation.



          Hope this helps






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            A toaster oven certainly uses radiation, but I'd argue that a bake-a-cake oven also uses conduction and convection.
            $endgroup$
            – psitae
            Dec 22 '18 at 20:17






          • 2




            $begingroup$
            @psitae What's your point? Whether its conduction, convection, radiation (microwave or infrared) none of these cook by exciting electrons.
            $endgroup$
            – Bob D
            Dec 22 '18 at 21:26










          • $begingroup$
            Why mention photons in this context?
            $endgroup$
            – Pieter
            Dec 23 '18 at 19:11














          3












          3








          3





          $begingroup$

          The microwave photon energy level gives rotational kinetic energy to water molecules due to interaction of the electromagnetic field with the water dipole molecule. The rotational kinetic energy is subsequently randomized to increase the average translational kinetic energy of the molecules, increasing the temperature of the water molecules. The microwave energy levels are well below those needed to excite electrons to higher levels.



          A regular oven cooks with infrared electromagnetic energy. The photon energy level corresponds to molecular vibration. These energy levels are likewise too low for electron excitation.



          Hope this helps






          share|cite|improve this answer









          $endgroup$



          The microwave photon energy level gives rotational kinetic energy to water molecules due to interaction of the electromagnetic field with the water dipole molecule. The rotational kinetic energy is subsequently randomized to increase the average translational kinetic energy of the molecules, increasing the temperature of the water molecules. The microwave energy levels are well below those needed to excite electrons to higher levels.



          A regular oven cooks with infrared electromagnetic energy. The photon energy level corresponds to molecular vibration. These energy levels are likewise too low for electron excitation.



          Hope this helps







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 22 '18 at 17:45









          Bob DBob D

          2,376212




          2,376212












          • $begingroup$
            A toaster oven certainly uses radiation, but I'd argue that a bake-a-cake oven also uses conduction and convection.
            $endgroup$
            – psitae
            Dec 22 '18 at 20:17






          • 2




            $begingroup$
            @psitae What's your point? Whether its conduction, convection, radiation (microwave or infrared) none of these cook by exciting electrons.
            $endgroup$
            – Bob D
            Dec 22 '18 at 21:26










          • $begingroup$
            Why mention photons in this context?
            $endgroup$
            – Pieter
            Dec 23 '18 at 19:11


















          • $begingroup$
            A toaster oven certainly uses radiation, but I'd argue that a bake-a-cake oven also uses conduction and convection.
            $endgroup$
            – psitae
            Dec 22 '18 at 20:17






          • 2




            $begingroup$
            @psitae What's your point? Whether its conduction, convection, radiation (microwave or infrared) none of these cook by exciting electrons.
            $endgroup$
            – Bob D
            Dec 22 '18 at 21:26










          • $begingroup$
            Why mention photons in this context?
            $endgroup$
            – Pieter
            Dec 23 '18 at 19:11
















          $begingroup$
          A toaster oven certainly uses radiation, but I'd argue that a bake-a-cake oven also uses conduction and convection.
          $endgroup$
          – psitae
          Dec 22 '18 at 20:17




          $begingroup$
          A toaster oven certainly uses radiation, but I'd argue that a bake-a-cake oven also uses conduction and convection.
          $endgroup$
          – psitae
          Dec 22 '18 at 20:17




          2




          2




          $begingroup$
          @psitae What's your point? Whether its conduction, convection, radiation (microwave or infrared) none of these cook by exciting electrons.
          $endgroup$
          – Bob D
          Dec 22 '18 at 21:26




          $begingroup$
          @psitae What's your point? Whether its conduction, convection, radiation (microwave or infrared) none of these cook by exciting electrons.
          $endgroup$
          – Bob D
          Dec 22 '18 at 21:26












          $begingroup$
          Why mention photons in this context?
          $endgroup$
          – Pieter
          Dec 23 '18 at 19:11




          $begingroup$
          Why mention photons in this context?
          $endgroup$
          – Pieter
          Dec 23 '18 at 19:11











          3












          $begingroup$

          On the case of a microwave oven the water molecules in the food are made to vibrate. Water molecules are tiny electric dipoles and these are made to vibrate more strongly. The molecules stay in the electronic ground state.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            What happens in ordinary ovens/burners? For heating to take place, a necessary condition is that the system must absorb energy. For this to happen, there must be a coupling between the environment that supplies energy and the system that absorbs it. For the microwave example, it is the electromagnetic microwave that couples to the dipole moment of water molecules.
            $endgroup$
            – mithusengupta123
            Dec 22 '18 at 17:14










          • $begingroup$
            The microwave photon energy level causes rotation not vibration of water dipole molecules. The vibration energy is higher than the microwave oven photon energy. For a good explanation of the interaction of radiation with matter, check out the Hyperphysics web site
            $endgroup$
            – Bob D
            Dec 22 '18 at 17:24












          • $begingroup$
            @BobD "Photon" does not explain anything here. The electric field polarizes the dipole orientation which relaxes when the field is switched off again. The field changes direction $2.45cdot10^9$ times per second, so that becomes noticeable as heat.
            $endgroup$
            – Pieter
            Dec 23 '18 at 19:17












          • $begingroup$
            @Pieter With all due respect, photon energy explains why microwave and infrared radiation do not excite electrons resulting in cooking, which I thought the OP was wondering. It also explains why the electric field in the kilohertz range does not cause molecule rotation.
            $endgroup$
            – Bob D
            Dec 24 '18 at 14:43


















          3












          $begingroup$

          On the case of a microwave oven the water molecules in the food are made to vibrate. Water molecules are tiny electric dipoles and these are made to vibrate more strongly. The molecules stay in the electronic ground state.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            What happens in ordinary ovens/burners? For heating to take place, a necessary condition is that the system must absorb energy. For this to happen, there must be a coupling between the environment that supplies energy and the system that absorbs it. For the microwave example, it is the electromagnetic microwave that couples to the dipole moment of water molecules.
            $endgroup$
            – mithusengupta123
            Dec 22 '18 at 17:14










          • $begingroup$
            The microwave photon energy level causes rotation not vibration of water dipole molecules. The vibration energy is higher than the microwave oven photon energy. For a good explanation of the interaction of radiation with matter, check out the Hyperphysics web site
            $endgroup$
            – Bob D
            Dec 22 '18 at 17:24












          • $begingroup$
            @BobD "Photon" does not explain anything here. The electric field polarizes the dipole orientation which relaxes when the field is switched off again. The field changes direction $2.45cdot10^9$ times per second, so that becomes noticeable as heat.
            $endgroup$
            – Pieter
            Dec 23 '18 at 19:17












          • $begingroup$
            @Pieter With all due respect, photon energy explains why microwave and infrared radiation do not excite electrons resulting in cooking, which I thought the OP was wondering. It also explains why the electric field in the kilohertz range does not cause molecule rotation.
            $endgroup$
            – Bob D
            Dec 24 '18 at 14:43
















          3












          3








          3





          $begingroup$

          On the case of a microwave oven the water molecules in the food are made to vibrate. Water molecules are tiny electric dipoles and these are made to vibrate more strongly. The molecules stay in the electronic ground state.






          share|cite|improve this answer











          $endgroup$



          On the case of a microwave oven the water molecules in the food are made to vibrate. Water molecules are tiny electric dipoles and these are made to vibrate more strongly. The molecules stay in the electronic ground state.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 22 '18 at 17:12

























          answered Dec 22 '18 at 16:22









          my2ctsmy2cts

          4,8462618




          4,8462618












          • $begingroup$
            What happens in ordinary ovens/burners? For heating to take place, a necessary condition is that the system must absorb energy. For this to happen, there must be a coupling between the environment that supplies energy and the system that absorbs it. For the microwave example, it is the electromagnetic microwave that couples to the dipole moment of water molecules.
            $endgroup$
            – mithusengupta123
            Dec 22 '18 at 17:14










          • $begingroup$
            The microwave photon energy level causes rotation not vibration of water dipole molecules. The vibration energy is higher than the microwave oven photon energy. For a good explanation of the interaction of radiation with matter, check out the Hyperphysics web site
            $endgroup$
            – Bob D
            Dec 22 '18 at 17:24












          • $begingroup$
            @BobD "Photon" does not explain anything here. The electric field polarizes the dipole orientation which relaxes when the field is switched off again. The field changes direction $2.45cdot10^9$ times per second, so that becomes noticeable as heat.
            $endgroup$
            – Pieter
            Dec 23 '18 at 19:17












          • $begingroup$
            @Pieter With all due respect, photon energy explains why microwave and infrared radiation do not excite electrons resulting in cooking, which I thought the OP was wondering. It also explains why the electric field in the kilohertz range does not cause molecule rotation.
            $endgroup$
            – Bob D
            Dec 24 '18 at 14:43




















          • $begingroup$
            What happens in ordinary ovens/burners? For heating to take place, a necessary condition is that the system must absorb energy. For this to happen, there must be a coupling between the environment that supplies energy and the system that absorbs it. For the microwave example, it is the electromagnetic microwave that couples to the dipole moment of water molecules.
            $endgroup$
            – mithusengupta123
            Dec 22 '18 at 17:14










          • $begingroup$
            The microwave photon energy level causes rotation not vibration of water dipole molecules. The vibration energy is higher than the microwave oven photon energy. For a good explanation of the interaction of radiation with matter, check out the Hyperphysics web site
            $endgroup$
            – Bob D
            Dec 22 '18 at 17:24












          • $begingroup$
            @BobD "Photon" does not explain anything here. The electric field polarizes the dipole orientation which relaxes when the field is switched off again. The field changes direction $2.45cdot10^9$ times per second, so that becomes noticeable as heat.
            $endgroup$
            – Pieter
            Dec 23 '18 at 19:17












          • $begingroup$
            @Pieter With all due respect, photon energy explains why microwave and infrared radiation do not excite electrons resulting in cooking, which I thought the OP was wondering. It also explains why the electric field in the kilohertz range does not cause molecule rotation.
            $endgroup$
            – Bob D
            Dec 24 '18 at 14:43


















          $begingroup$
          What happens in ordinary ovens/burners? For heating to take place, a necessary condition is that the system must absorb energy. For this to happen, there must be a coupling between the environment that supplies energy and the system that absorbs it. For the microwave example, it is the electromagnetic microwave that couples to the dipole moment of water molecules.
          $endgroup$
          – mithusengupta123
          Dec 22 '18 at 17:14




          $begingroup$
          What happens in ordinary ovens/burners? For heating to take place, a necessary condition is that the system must absorb energy. For this to happen, there must be a coupling between the environment that supplies energy and the system that absorbs it. For the microwave example, it is the electromagnetic microwave that couples to the dipole moment of water molecules.
          $endgroup$
          – mithusengupta123
          Dec 22 '18 at 17:14












          $begingroup$
          The microwave photon energy level causes rotation not vibration of water dipole molecules. The vibration energy is higher than the microwave oven photon energy. For a good explanation of the interaction of radiation with matter, check out the Hyperphysics web site
          $endgroup$
          – Bob D
          Dec 22 '18 at 17:24






          $begingroup$
          The microwave photon energy level causes rotation not vibration of water dipole molecules. The vibration energy is higher than the microwave oven photon energy. For a good explanation of the interaction of radiation with matter, check out the Hyperphysics web site
          $endgroup$
          – Bob D
          Dec 22 '18 at 17:24














          $begingroup$
          @BobD "Photon" does not explain anything here. The electric field polarizes the dipole orientation which relaxes when the field is switched off again. The field changes direction $2.45cdot10^9$ times per second, so that becomes noticeable as heat.
          $endgroup$
          – Pieter
          Dec 23 '18 at 19:17






          $begingroup$
          @BobD "Photon" does not explain anything here. The electric field polarizes the dipole orientation which relaxes when the field is switched off again. The field changes direction $2.45cdot10^9$ times per second, so that becomes noticeable as heat.
          $endgroup$
          – Pieter
          Dec 23 '18 at 19:17














          $begingroup$
          @Pieter With all due respect, photon energy explains why microwave and infrared radiation do not excite electrons resulting in cooking, which I thought the OP was wondering. It also explains why the electric field in the kilohertz range does not cause molecule rotation.
          $endgroup$
          – Bob D
          Dec 24 '18 at 14:43






          $begingroup$
          @Pieter With all due respect, photon energy explains why microwave and infrared radiation do not excite electrons resulting in cooking, which I thought the OP was wondering. It also explains why the electric field in the kilohertz range does not cause molecule rotation.
          $endgroup$
          – Bob D
          Dec 24 '18 at 14:43













          0












          $begingroup$

          The heat contained in a system can be thought of as the total kinetic energy of a system. my2cts' answer is correct, but I'll add that if you excite an electron to a higher energy level, that will indeed cause heating in many cases (see for example FRET).



          The friction between the water molecules and the rest of the system is what causes things to heat up in a microwave. In a conventional oven, it's a direct transfer of kinetic energy from hot air.



          Homework question: would something heat up in a microwave if it didn't contain any water?






          share|cite|improve this answer









          $endgroup$









          • 2




            $begingroup$
            Heat is energy transfer from one substance to another solely due to a temperature difference between them. Systems do not "contain" heat. You cannot think of heat as the total kinetic energy of a system. The proper term for the total kinetic energy (plus potential energy) is internal energy.
            $endgroup$
            – Bob D
            Dec 22 '18 at 21:31












          • $begingroup$
            @BobD that's correct. I'll edit my answer.
            $endgroup$
            – psitae
            Dec 22 '18 at 21:34
















          0












          $begingroup$

          The heat contained in a system can be thought of as the total kinetic energy of a system. my2cts' answer is correct, but I'll add that if you excite an electron to a higher energy level, that will indeed cause heating in many cases (see for example FRET).



          The friction between the water molecules and the rest of the system is what causes things to heat up in a microwave. In a conventional oven, it's a direct transfer of kinetic energy from hot air.



          Homework question: would something heat up in a microwave if it didn't contain any water?






          share|cite|improve this answer









          $endgroup$









          • 2




            $begingroup$
            Heat is energy transfer from one substance to another solely due to a temperature difference between them. Systems do not "contain" heat. You cannot think of heat as the total kinetic energy of a system. The proper term for the total kinetic energy (plus potential energy) is internal energy.
            $endgroup$
            – Bob D
            Dec 22 '18 at 21:31












          • $begingroup$
            @BobD that's correct. I'll edit my answer.
            $endgroup$
            – psitae
            Dec 22 '18 at 21:34














          0












          0








          0





          $begingroup$

          The heat contained in a system can be thought of as the total kinetic energy of a system. my2cts' answer is correct, but I'll add that if you excite an electron to a higher energy level, that will indeed cause heating in many cases (see for example FRET).



          The friction between the water molecules and the rest of the system is what causes things to heat up in a microwave. In a conventional oven, it's a direct transfer of kinetic energy from hot air.



          Homework question: would something heat up in a microwave if it didn't contain any water?






          share|cite|improve this answer









          $endgroup$



          The heat contained in a system can be thought of as the total kinetic energy of a system. my2cts' answer is correct, but I'll add that if you excite an electron to a higher energy level, that will indeed cause heating in many cases (see for example FRET).



          The friction between the water molecules and the rest of the system is what causes things to heat up in a microwave. In a conventional oven, it's a direct transfer of kinetic energy from hot air.



          Homework question: would something heat up in a microwave if it didn't contain any water?







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 22 '18 at 17:18









          psitaepsitae

          645526




          645526








          • 2




            $begingroup$
            Heat is energy transfer from one substance to another solely due to a temperature difference between them. Systems do not "contain" heat. You cannot think of heat as the total kinetic energy of a system. The proper term for the total kinetic energy (plus potential energy) is internal energy.
            $endgroup$
            – Bob D
            Dec 22 '18 at 21:31












          • $begingroup$
            @BobD that's correct. I'll edit my answer.
            $endgroup$
            – psitae
            Dec 22 '18 at 21:34














          • 2




            $begingroup$
            Heat is energy transfer from one substance to another solely due to a temperature difference between them. Systems do not "contain" heat. You cannot think of heat as the total kinetic energy of a system. The proper term for the total kinetic energy (plus potential energy) is internal energy.
            $endgroup$
            – Bob D
            Dec 22 '18 at 21:31












          • $begingroup$
            @BobD that's correct. I'll edit my answer.
            $endgroup$
            – psitae
            Dec 22 '18 at 21:34








          2




          2




          $begingroup$
          Heat is energy transfer from one substance to another solely due to a temperature difference between them. Systems do not "contain" heat. You cannot think of heat as the total kinetic energy of a system. The proper term for the total kinetic energy (plus potential energy) is internal energy.
          $endgroup$
          – Bob D
          Dec 22 '18 at 21:31






          $begingroup$
          Heat is energy transfer from one substance to another solely due to a temperature difference between them. Systems do not "contain" heat. You cannot think of heat as the total kinetic energy of a system. The proper term for the total kinetic energy (plus potential energy) is internal energy.
          $endgroup$
          – Bob D
          Dec 22 '18 at 21:31














          $begingroup$
          @BobD that's correct. I'll edit my answer.
          $endgroup$
          – psitae
          Dec 22 '18 at 21:34




          $begingroup$
          @BobD that's correct. I'll edit my answer.
          $endgroup$
          – psitae
          Dec 22 '18 at 21:34


















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