Question

For isothermal expansion case of an ideal gas, the correct combination of the thermodynamic parameters will be:
A.\[\Delta \;U = 0{\text{ }},{\text{ }}Q = 0,{\text{ }}W \ne 0{\text{ and }}\Delta H \ne 0\]
B.\[\Delta \;U \ne 0{\text{ }},{\text{ }}Q \ne 0,{\text{ }}W \ne 0{\text{ and }}\Delta H = 0\]
C.\[\Delta \;U = 0{\text{ }},{\text{ }}Q \ne 0,{\text{ }}W = 0{\text{ and }}\Delta H \ne 0\]
D.\[\Delta \;U = 0{\text{ }},{\text{ }}Q \ne 0,{\text{ }}W \ne 0{\text{ and }}\Delta H = 0\]

Answer 1

Hint: Isothermal refers to no change in temperature and apply this fact on different energies in the aforementioned options i.e. internal energy, work done, the heat of reaction and enthalpy of reaction.

Complete step by step answer:
The term “isothermal expansion” means an increase in the volume of the gas at a constant temperature.
The relations we need to find are: change in internal energy ($\Delta U$), work done ($W$), heat of reaction ($Q$) and change in enthalpy ($\Delta H$).
Now let us analyse each variable systematically.
Internal energy is a function of temperature because internal energy of ideal gas includes molecular kinetic energy which is dependent on the temperature and hence, for an isothermal process change in temperature is zero. Thus, $\Delta U = 0$.
As the temperature remains constant and gas is expanding, it means the molecules of the gas are moving apart, which implies that negative work is being done on the system. This means that the value of $W$ will be a non-zero negative integer. Thus, $W \ne 0$.
We can write the equation for internal energy ($U$) as, $U = Q + W$.
From the aforementioned facts, we have concluded that $\Delta U = 0$ and $W$ is a non-zero negative number.
Using this information, we get,
 $
  U = Q + W \\
  0 = Q + W \\
  Q = - W \\
$
As $W$ is a non-zero negative number, it can be corroborated that $Q$ will be positive.
Hence, $Q \ne 0$.
Now, change in enthalpy $\Delta H$ can be written as \[\Delta H = n{C_p}\Delta T\] where,
$n$= no. of moles
${C_p}$= molar specific heat at constant pressure
$\Delta T$= change in temperature
As the given reaction is an isothermal reaction, $\Delta T = 0$.
Hence, $\Delta H = 0$.
So, we can say that $\Delta U = 0$, $Q \ne 0$, $W \ne 0$ and $\Delta H = 0$.

Therefore, we can conclude that the correct answer to this question is option D.

Note:
An isobaric process is one where the pressure of the system (often a gas) stays constant. An isochoric process is one where the volume of the system stays constant. An adiabatic process is a thermodynamic process in which no heat is exchanged between the system and the surrounding.