Question 209: Thermodynamics & KTG

Question

An ideal gas goes from an initial state to final state. During the process, the pressure of gas increases linearly with temperature.
A. The work done by gas during the process is zero.
B. The heat added to gas is different from change in its internal energy.
C. The volume of the gas is increased.
D. The internal energy of the gas is increased.
E. The process is isochoric (constant volume process)
Choose the correct answer from the options given below:

Accepted Answer

For an ideal gas, the equation of state is given by
$$ pV = nRT. $$
If during a process the pressure increases linearly with temperature (i.e., $$p \propto T$$), then from the ideal gas law (with a fixed amount of gas, $$n$$, and with the gas constant, $$R$$), it follows that
$$ \frac{p}{T} = \frac{nR}{V}. $$
Since $$nR$$ is constant, a linear relationship between $$p$$ and $$T$$ implies that $$\frac{p}{T}$$ remains constant throughout the process. This can only be true if the volume $$V$$ is constant. Hence, the process is isochoric.
For an isochoric (constant volume) process:
The work done by the gas is given by
$$   W = \int p\, dV,   $$
and since $$dV = 0$$, we have $$W = 0.$$ (Statement A is true.)
The internal energy change for an ideal gas depends only on temperature:
$$   \Delta U = nC_V\Delta T.   $$
If the temperature increases, then $$\Delta U > 0.$$ (Statement D is true.)
For an isochoric process, no expansion work is done, so any heat added goes entirely into increasing the internal energy. This means the heat added is not different from the change in internal energy (in fact, $$Q = \Delta U$$). (Statement B is false.)
Since the process is isochoric, the volume does not change (and certainly is not increased). (Statement C is false, and Statement E stating it is isochoric is true.)
Thus, the correct true statements are A, D, and E.

Answer

A, C Only

Answer

A, D, E Only

Answer

E Only

Answer

A, B, C, D Only

Thermodynamics & KTG

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