# Thermodynamics: Comment on mistake

## Homework Statement

Suppose you have a mole of water at 25##^{\circ}## at atmospheric pressure. You then raise its temperature to 30##^{\circ}##. Determine the change in Gibbs free energy. To compensate for the change, once can raise the pressure on water, how much pressure you need to increase?

I know the right way to do this problem, but I would like to know what I did wrong initially with the following steps:

I knew ##\Delta G = \Delta H - T\Delta S = \Delta Q - \Delta W_{other}##, so i thought: To raise temperature I'd probably need to provide heat, and since no mention of any non-mechanical work was made in the problem, let's take ##\Delta W_{other} = 0##

Therefore ##\Delta G = \Delta Q = C_p \Delta T##, where ##C_p## was a given.

I would then use the thermodynamic identity to work out the pressure change.

What were the wrong assumptions made?

Assistance is greatly appreciated

## The Attempt at a Solution

BvU
Homework Helper
Hi,

In my book ##\Delta H = c_p \Delta T##, so you missed the ##T\Delta S## altogether ? Anyway, if T is not a constant, what do you do with a term like ##T\Delta S## ?

Hi,

In my book ##\Delta H = c_p \Delta T##, so you missed the ##T\Delta S## altogether ?

Good heavens, I just spotted my mistake. It should have been ##\Delta G = \Delta W_{other}## at the end when ##Q## and ##T\Delta S## cancel. Thanks for pointing it out.

I guess that even this result wouldn't have been the most useful.

I would have made use of the fact that ##\Delta G / \Delta T = -S## to solve the problem by using the ##G## thermo-identity.

Hi,
Anyway, if T is not a constant, what do you do with a term like ##T\Delta S## ?

Could you elaborate on what you mean by this? Apologies.

BvU
Homework Helper
Could you elaborate
My comment was exclusively triggered by your ##
\Delta G = \Delta H - T\Delta S = \Delta Q - \Delta W_{other}##.

However, the natural variables are ##T,p## and ##{N_i}## so that ##dG = -SdT + V dP +\sum\mu_idN_i## or ##\Delta G = -S\Delta T## as you found for your isobaric scenario.

Chestermiller
Mentor
Is it possible you are supposed to solve this problem using thermodynamic tables? Otherwise, what is your relationship for ##\Delta S## at constant pressure?

BvU
Is it possible you are supposed to solve this problem using thermodynamic tables? Otherwise, what is your relationship for ##\Delta S## at constant pressure?

If what is meant by thermodynamic tables is given values for quantities like entropy then yes, I was. I guess most of my confusion came from not knowing where to start, given the table of values.

Chestermiller
Mentor
If what is meant by thermodynamic tables is given values for quantities like entropy then yes, I was. I guess most of my confusion came from not knowing where to start, given the table of values.
Does your (compressed water) table give values for P, T, v, u, h, and s?