Question

For isothermal reversible expansion process, the value of \[q\] can be calculated by the expression
A.\[q = 2.303nRT{\text{log}}\left( {\dfrac{{{V_{2}}}}{{{V_1}}}} \right)\]
B.\[q = - 2.30nRT{\text{log}}\left( {\dfrac{{{V_{2}}}}{{{V_1}}}} \right)\]
C.\[q = - 2.303nRT{\text{log}}\left( {\dfrac{{{V_{2}}}}{{{V_1}}}} \right)\]
D.\[q = - {P_{ext}}nRT{\text{log}}\left( {\dfrac{{{V_{2}}}}{{{V_1}}}} \right)\]

Answer 1

Hint: To answer this question, you should recall the concept of isothermal process. An isothermal process is a thermodynamic process in which the temperature of a system remains constant. We shall substitute the values given in the question and compare it with the equations of the three laws of thermodynamics and then with work done for reversible expansion.

Complete Step by step solution:
In an ideal gas, all the collisions between molecules or atoms are perfectly elastic and no intermolecular force of attraction exists in an ideal gas because molecules of an ideal gas move so fast, and they are so far away from each other that they do not interact at all. In the case of real gas, they have negligible intermolecular attractive forces.
For an isothermal reversible process, \[\Delta T = 0\;\]. Also change in internal energy i.e. \[\Delta U = n{C_V}\Delta T = 0\].
We know from the first law of thermodynamics that \[\;\Delta U = q + w = 0\].
\[\therefore \] Using the properties of a reversible isothermal expansion we can say:
\[q = - w\].
For an isothermal reversible expansion,
\[w = - nRT{\text{ln}}\left( {\dfrac{{{V_2}}}{{{V_1}}}} \right) = - 2.303nRT{\text{log}}\left( {\dfrac{{{V_2}}}{{{V_1}}}} \right)\]
Substituting:
\[\therefore q = - w = 2.303nRT{\text{log}}\left( {\dfrac{{{V_{2}}}}{{{V_1}}}} \right)\]

Hence, the correct option is A.

Note: We need to know about each of the thermodynamic processes mentioned in the aforementioned question.
- An isothermal process is a type of thermodynamic process in which the temperature change is zero. This is achieved when transfer of heat into or out of the system takes place very slowly such that thermal equilibrium is maintained.
- An adiabatic process is the type of thermodynamic process in which there is no exchange of heat from the system to its surrounding. The heat exchange is zero both during expansion and compression.
- An Isobaric process is the type of thermodynamic process where pressure change stays constant.
- An Isochoric process is the type of thermodynamic process where the volume of the system remains constant, is called an isochoric process. For example: Heating of gas in a closed cylinder is an isochoric process.