Which of the following correctly represents the formula for the molar heat capacity at constant pressure, c_p?

• A. c_p = q/n
• B. c_p = (∂H/∂T)_p
• C. c_p = ΔH/ΔP
• D. c_p = (∂U/∂T)_p

Answer: (B)

Verified step by step guidance

1. Understand that molar heat capacity at constant pressure, denoted as \(c_p\), is defined as the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius (or Kelvin) while keeping the pressure constant.
2. Recall that heat added at constant pressure is related to the change in enthalpy \(H\), so the molar heat capacity at constant pressure can be expressed as the rate of change of enthalpy with respect to temperature at constant pressure.
3. Write the mathematical definition of molar heat capacity at constant pressure as \(c_p = \left(\frac{\partial H}{\partial T}\right)_p\), where the partial derivative indicates that pressure is held constant during the temperature change.
4. Recognize that \(c_p = \frac{q}{n}\) is incomplete because \(q\) must be the heat added at constant pressure and the temperature change must be considered; the correct expression involves the derivative of enthalpy with respect to temperature, not just heat per mole.
5. Note that expressions involving changes in pressure \(\Delta P\) or internal energy \(U\) at constant pressure are incorrect for \(c_p\), since \(c_p\) specifically relates to enthalpy changes with temperature at constant pressure.