Question

Assertion : $ {C_p} - {C_v} = R $ for an ideal gas.
Reason : $ {\left[ {\dfrac{{\partial U}}{{\partial V}}} \right]_T} = 0 $ for an ideal gas.
(A) Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
(B) Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
(C) Assertion is correct but Reason is incorrect
(D) Assertion is incorrect but Reason is correct

Answer 1

Hint: Ideal gas is a hypothetical gas composed of many randomly moving point particles. Ideal gas obeys ideal gas law. We shall relate the 1st law of thermodynamics with heat capacity to find the relation required.

Formula Used: $ \Delta H = \Delta U + W $
 $ \Delta H = n{C_P}\Delta T $
 $ \Delta U = n{C_V}\Delta T $
where, H is quantity of energy supplied as heat, U denotes the change in internal energy, W denotes the amount of thermodynamic work done by the system, $ {C_p} $ is the specific heat capacity at constant pressure $ {C_v} $ is specific heat capacity at constant volume and T is the temperature.

Complete step by step solution:
C is the specific heat capacity of a substance. Specific heat capacity of a substance is the heat capacity of a substance divided by mass of the sample. $ {C_p} $ is the specific heat capacity at constant pressure and $ {C_v} $ is specific heat capacity at constant volume. The SI unit of specific heat is Joule per Kelvin and kilogram or J/Kkg.
First law of thermodynamics relates to the law of conservation of energy. First law of thermodynamics relates heat, work, and internal energy. First law of thermodynamics states that energy can be transformed from one form to another, but it can neither be created nor destroyed. It is given by the following formula,
 $ \Delta H = \Delta U + W $
where, H is quantity of energy supplied as heat, U denotes the change in internal energy and W denotes the amount of thermodynamic work done by the system.
 For an ideal gas, H is given by
 $ \Delta H = n{C_P}\Delta T $
U is given by,
 $ \Delta U = n{C_V}\Delta T $
W is given by,
W= $ \Delta PV $
where P is pressure, V is volume, n is the number of moles, R is universal gas constant, T is the temperature.
Hence, first law of thermodynamics can be given by,
 $ n{C_P}\Delta T = n{C_V}\Delta T + \Delta PV $
 $ \Delta PV = nR\Delta T $
 $ n{C_P}\Delta T = n{C_V}\Delta T + nR\Delta T $
 $ {C_P} - {C_V} = R $
Hence, the assertion is correct.
Internal energy depends on temperature. Hence, change with volume does not affect internal energy at constant temperature.
For an ideal gas,
 $ {\left[ {\dfrac{{dU}}{{dV}}} \right]_T} = 0 $
Hence the reason is also correct.
Both assertion and reason are correct, but reason is not the correct explanation for assertion.
Therefore, the correct answer is option B.

Note:
Real gas does not obey the ideal gas equation. It obeys Vander Waals equation. The real gas equation can be converted to the ideal gas situation by assuming negligible intermolecular forces and volume occupied.