Question

Which is not a correct relationship?
(A)- \[{{\left[ \dfrac{dH}{dT} \right]}_{P}}-{{\left[ \dfrac{dU}{dT} \right]}_{V}}=-ve\]
(B)- \[{{\left[ \dfrac{dU}{dV} \right]}_{T}}=0\] for an ideal gas
(C)- \[{{\left[ \dfrac{dV}{dT} \right]}_{_{P}}}=\dfrac{nR}{P}\] for an ideal gas
(D)- None

Answer 1

Hint: Cp is the heat capacity at constant pressure and is equal to the rate of change of enthalpy at constant pressure. Cv is the heat capacity at constant volume and is defined at the rate of change of energy at constant volume.


Complete step by step solution:
Let's look why some options are incorrect.
-Energy (U) is a function of temperature. It id dependent on the temperature of the gas. If change in temperature is 0 then change in energy will also be 0. So, for an ideal gas \[{{\left[ \dfrac{dU}{dV} \right]}_{T}}=0\]is true. Hence OPTION (B) is incorrect as we have to choose the incorrect statement.
-From the ideal gas equation, we get,
\[PV=nRT\]
Now, to get the desired result we will differentiate both sides
\[PdV+VdP=nRdT\]
Now, at constant pressure, dp=0
Therefore, \[P{{(dV)}_{P}}=nR{{(dT)}_{P}}\]
\[{{\left[ \dfrac{dV}{dT} \right]}_{_{P}}}=\dfrac{nR}{P}\] for an ideal gas
So, the above result is true for an ideal gas.
Hence, OPTION (C) is incorrect as we have to choose the false statement.
- Cp is the heat capacity at constant pressure and is equal to the rate of change of enthalpy at constant pressure. Cv is the heat capacity at constant volume and is defined at the rate of change of energy at constant volume.
Therefore, Cp \[={{\left[ \dfrac{dH}{dT} \right]}_{P}}\]and Cv \[={{\left[ \dfrac{dU}{dT} \right]}_{V}}\]
Now, we know that Cp is greater than Cv. So the difference of them should be positive but in the OPTION (A) it is given that,
\[{{\left[ \dfrac{dH}{dT} \right]}_{P}}-{{\left[ \dfrac{dU}{dT} \right]}_{V}}= -ve\] which is an incorrect statement.

Hence option (A) is the correct answer to the above question.

Note: Cp is greater than Cv as when heat is supplied at constant volume then no work is done, only temperature is increased but when heat is supplied at constant pressure then work is done also.