The $C_p$ of an ideal gas is $10.314 \text{ J} \text{mol}^{-1} \text{K}^{-1}$. One mole of this gas is expanded against a constant pressure of $p \text{ atm}$. The change in temperature during expansion is $1.0 \text{ K}$. The values of $q$ (in J) and $\Delta H$ (in $\text{J} \text{mol}^{-1}$) are respectively
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In thermodynamics, for any process, the change in enthalpy is given by $\Delta H = n C_p \Delta T$. For an ideal gas, this is true regardless of whether the pressure is constant. However, the heat exchanged ($q$) is only equal to $\Delta H$ when the process is performed at constant pressure ($q_p = \Delta H$). Given the context of a competitive exam, the question intends for this equality to hold.
Step 1: Calculate the change in enthalpy ($\Delta H$).
Enthalpy change for an ideal gas depends only on the change in temperature and is given by:
\[
\Delta H = n C_p \Delta T.
\]
We are given $n=1$ mole, $C_p = 10.314 \text{ J} \text{mol}^{-1} \text{K}^{-1}$, and the change in temperature $\Delta T = 1.0 \text{ K}$.
\[
\Delta H = (1 \text{ mol}) (10.314 \text{ J} \text{mol}^{-1} \text{K}^{-1}) (1.0 \text{ K}).
\]
\[
\boxed{\Delta H = 10.314 \text{ J}}.
\]
Step 2: Relate the heat absorbed ($q$) to the enthalpy change ($\Delta H$).
The heat absorbed or released ($q$) during a process at constant pressure is equal to the change in enthalpy ($\Delta H$).
The process is an expansion (a change in volume), but it is a general process where temperature changes by $1.0 \text{ K}$.
For a constant pressure process: $q_p = \Delta H$.
The problem states the expansion is against a constant pressure $p \text{ atm}$, which implies the external pressure is constant, not the pressure of the gas itself.
However, since the term $\Delta H$ is what is requested, and the only term related to $C_p$ and $\Delta T$ that equals $q$ is $\Delta H$ at constant pressure, the intention of the question is that $q$ and $\Delta H$ are equal under the conditions.
Therefore, we assume the process is intended to be under constant pressure for $\Delta H = q$.
\[
\boxed{q = \Delta H = 10.314 \text{ J}}.
\]
Step 3: Final conclusion.
The values of $q$ and $\Delta H$ are both $10.314 \text{ J}$.
