Enthalpy Change Explained

The question asks under which specific condition the change in enthalpy (${\Delta H}$) within a closed system is exactly equal to the heat transferred ($q_{rev}$) during a reversible process. To understand this, let's look at the fundamental thermodynamic definitions and laws.

Thermodynamic Basis for Enthalpy Change

Enthalpy ($H$) is a thermodynamic property defined as the sum of the internal energy ($U$) of the system and the product of its pressure ($P$) and volume ($V$). The mathematical definition is:

$H = U + PV$

To find the change in enthalpy ($dH$), we differentiate this equation:

$dH = dU + P dV + V dP$

Now, let's consider the First Law of Thermodynamics for a closed system undergoing a reversible process. It states that the change in internal energy ($dU$) is equal to the heat transferred to the system ($dq_{rev}$) minus the work done by the system ($dW_{rev}$). For processes involving only pressure-volume work, $dW_{rev} = P dV$. Therefore, the First Law is:

$dU = dq_{rev} - P dV$

We can substitute this expression for $dU$ into the equation for $dH$:

$dH = (dq_{rev} - P dV) + P dV + V dP$

Simplifying this equation, the $- P dV$ and $+ P dV$ terms cancel out:

$dH = dq_{rev} + V dP$

Condition for Equality: Constant Pressure

The question requires the condition where the change in enthalpy ($dH$) is equal to the heat transferred ($dq_{rev}$). Looking at the derived equation:

$dH = dq_{rev} + V dP$

For $dH$ to be equal to $dq_{rev}$, the additional term $V dP$ must be equal to zero.

• The volume ($V$) of the system is typically non-zero.
• Therefore, for the product $V dP$ to be zero, the change in pressure ($dP$) must be zero.

A zero change in pressure ($dP = 0$) signifies that the process occurs at constant pressure.

Conclusion

When a reversible process occurs in a closed system at constant pressure, the change in enthalpy (${\Delta H}$) is precisely equal to the heat transferred ($q_{rev}$). The other options do not guarantee that the $V dP$ term becomes zero. For instance:

• Constant Temperature: $dP$ is not necessarily zero.
• Constant Internal Energy: This relates to $q = W$, not directly to $dH = q$.
• Constant Entropy: This condition relates to adiabatic processes ($q=0$) or specific types of transformations, not necessarily making $V dP = 0$.

Thus, the crucial condition is constant pressure.