Question

The value of $\Delta H$ for cooling $2$ moles of an ideal monatomic gas from ${225^ \circ }C$ to ${125^ \circ }C$ at constant pressure will be:
Given ${C_p} = \dfrac{5}{2}R$
A.$250R$
B.$- 500R$
C.$500R$
D.$- 250R$

Answer 1

We know that the value of $\Delta H$ is calculated as $- n{C_p}\Delta T$ where $n$ is number of moles of the gas, ${C_p}$ is the heat capacity at constant pressure and $\Delta T$ is the temperature difference at constant pressure.

Complete step by step solution:
First of all let us read about heat capacity at constant pressure and heat capacity at constant volume.
Heat capacity at constant pressure: It is defined as the amount of heat energy absorbed or released by the substance with the change in temperature at constant pressure. It is represented as ${C_p}$.
Heat capacity at constant volume: It is defined as the amount of heat energy absorbed or released by the substance with the change in temperature at constant volume. It is represented as ${C_v}$.
Ideal gas: The gases which follow the ideal gas relation i.e. $PV = nRT$, where $P$ is the pressure, $V$ is the volume, $n$ is the number of moles of gas, $T$ is the temperature and $R$ is gas constant, in all conditions, are known as ideal gases.
Relation between ${C_p}$ and ${C_v}$ for the ideal gas is as: ${C_p} - {C_v} = R$.
Enthalpy: It is defined as the sum of internal energy and the product of pressure and volume, is known as enthalpy. It is represented by $H$. So according to the definition the enthalpy will be $H = U + PV$ where $U$ is internal energy, $P$ is the pressure and $V$ is the volume.
Change in enthalpy: It is defined as the amount of heat energy absorbed or released by the substance at the constant pressure. And it is calculated as : $\Delta H = - n{C_p}\Delta T$ where $n$ is number of moles of the gas, ${C_p}$ is the heat capacity at constant pressure and $\Delta T$ is the temperature difference at constant pressure. In the question it is given that the value of ${C_p}$ as $\dfrac{5}{2}R$ and $\Delta T$ is $225 - 125 = {100^ \circ }C$ and the value of $n$ as $2$. Hence the value of ${C_p}$ will be: $- 2 \times \dfrac{5}{2}R \times 100 = - 500R$.

So, option B is the correct.

Note: Internal energy of a system: It is defined as the energy associated with the random movement of the molecules, is known as the internal energy of a system. It is represented by the symbol $U$.
Change in internal energy: It is defined as the sum of the heat transferred and the work done.