Clapeyron equation: dp/dT = ΔS / ΔV

My textbook says, for fusion/melting reactions: dp/dT = ΔH / (T ΔV)

this implies that T * ΔS = ΔH.

I know T * ΔS = q and at constant pressure, that equals H, but the dp/dT reaction is not at constant pressure. Please explain. Thanks!

The Clapeyron equation you mentioned, dp/dT = ΔS / ΔV, is a general equation that relates the change in pressure (dp) to the change in temperature (dT) for any process occurring in a closed system. It is derived by applying the first law of thermodynamics to the system.

However, for fusion/melting reactions, where a substance undergoes a phase change from solid to liquid (or vice versa), a modified version of the Clapeyron equation is commonly used.

The equation you provided, dp/dT = ΔH / (T * ΔV), is a specific form of the Clapeyron equation that applies to fusion/melting reactions. It takes into account two additional factors: the enthalpy change (ΔH) and the change in volume (ΔV) associated with the phase transition.

Now, let's address your question about T * ΔS = ΔH. You are correct that T * ΔS represents the change in entropy (ΔS) multiplied by the temperature (T).

In a fusion/melting reaction at constant pressure, the enthalpy change (ΔH) is equivalent to the heat transferred (q) because we can assume that the only form of work done is expansion work (PΔV) and no other forms of work are involved.

So, in this case, ΔH = q, and since at constant pressure, q is equal to the change in enthalpy (H), we have ΔH = H.

Therefore, T * ΔS = ΔH holds true for a fusion/melting reaction at constant pressure.

However, it's important to note that the Clapeyron equation for fusion/melting reactions, dp/dT = ΔH / (T * ΔV), is derived treating the reaction within the framework of the idealized circumstances of a constant pressure process. Although the pressure may not be strictly constant during the phase transition, this equation is still a good approximation to describe the behavior of the system.