# How is Gibbs free energy measured?

##### 1 Answer

*It's not measured. Actually, other MEASURABLE variables (volume and pressure) are measured in a certain way so that we can calculate #DeltaG#.*

#color(blue)(DeltaG = int_(P_1)^(P_2) VdP)#

For the same-volume sample, change the pressure while keeping the temperature constant, and you can calculate

*DISCLAIMER: This can be a difficult topic, so ask questions if you are confused.*

If we take a look at this "thermodynamic square", we see the following terms:

#S# for entropy, a**natural variable**, placed on a**corner**.#H# for enthalpy, a state function, placed on a side.#P# for pressure, a**natural variable**, placed on a**corner**.#U# for internal energy, a state function, placed on a side.#G# for Gibbs' Free Energy, a state function, placed on a side.#V# for volume, a**natural variable**, placed on a**corner**.#A# for Helmholtz Free Energy, a state function, placed on a side.#T# for temperature, a**natural variable**, placed on a**corner**.

From these we can derive the **Maxwell Thermodynamics Relations**.

Since I couldn't find this square anywhere, I actually had to draw it myself. So, let's derive a familiar relationship to show that this square works.

**ENTHALPY VS. TEMPERATURE, ENTROPY, VOLUME, AND PRESSURE**

Let's use this square to find a Maxwell relation for **enthalpy**.

Because one arrow points from the side of ** towards** the temperature, which is a

**natural variable**, we write

**positive**

**volume**is a natural variable, and the other arrow also points from the side of

**the volume, so we write**

*towards***positive**

The remaining terms placed right next to **changing**. For those we write

So, we write:

#color(green)(dH = TdS + VdP)#

**GIBBS FREE ENERGY VS. ENTROPY, TEMPERATURE, VOLUME, AND PRESSURE**

Next, let's get a Maxwell relation for the **Gibbs' Free Energy**.

Since the arrow is pointing ** towards** temperature AND coming from the side

*opposite*to

**negative**. Entropy IS a natural variable, which is why it said to not be changing in this case. Likewise, volume is a natural variable, and it is being pointed

**, making it positive.**

*towards*Since *the same side* as

Therefore, we write:

#color(green)(dG = -SdT + VdP)#

I promise there's a point to this.

**THE FAMILIAR THERMODYNAMICS EQUATION**

So, let's compare **both contain**

#dG = -SdT + stackrel(dH)(overbrace(VdP + TdS)) - TdS#

#= -(SdT + TdS) + dH#

Hm... Familiar. The first term is a result of a product rule.

#dG = dH - d(TS)#

Or, as a more familiar notation:

#color(green)(DeltaG = DeltaH - Delta(TS))#

Or, at a constant temperature, like we normally use this equation in this form:

#DeltaG = DeltaH cancel(- SDeltaT) + TDeltaS#

#\mathbf(DeltaG = DeltaH - TDeltaS) larr# FAMILIAR!

And you should know this equation! That means both equations we derived are correct.

**HOW TO GET GIBBS' FREE ENERGY FROM MEASUREMENTS OF MEASURABLE VARIABLES**

Now that we've established that, let's go back to the Gibbs' free energy Maxwell relation.

#dG = -SdT + VdP#

Let's say we wanted to focus on the **partial derivative with respect to pressure at a constant temperature** to do so. Thus, we can say

#color(green)(((delG)/(delP))_T = V)#

Next, at a **constant temperature**, let us find an expression to use volume and pressure (two ** MEASURABLE** variables) to calculate the Gibbs' Free Energy:

#dG = VdP#

#color(blue)(DeltaG = int_(P_1)^(P_2) VdP)#

*Therefore, if we know the volume of the sample, and we change the pressure while keeping the experiment temperature constant, we can calculate #DeltaG# using measurable variables.*