Calculate the change in molar Gibbs energy of carbon dioxide gas at `20^(@)` C when it is isothermally compressed from 1.0 bar to 2.0 bar.

Options

A. 2.4 kJ/mol
B. `-2.4` kJ/mol
C. `1.7 kJ//mol`
D. `-1.7` kJ/mol

Explanation

To calculate the change in molar Gibbs energy (ΔG) of carbon dioxide gas when it is isothermally compressed from 1.0 bar to 2.0 bar at 20°C, we can follow these steps:

### Step 1: Understand the Conditions
We are given that the process is isothermal, meaning the temperature (T) remains constant. The initial pressure (P_initial) is 1.0 bar, and the final pressure (P_final) is 2.0 bar. The temperature is given as 20°C, which we need to convert to Kelvin.

**Hint:** Remember that temperature in Kelvin is calculated as \( T(K) = T(°C) + 273.15 \).

### Step 2: Convert Temperature to Kelvin
Convert 20°C to Kelvin:
\[ T = 20 + 273.15 = 293.15 \, K \]

### Step 3: Use the Gibbs Energy Change Formula
The change in Gibbs energy for an ideal gas under isothermal conditions can be expressed as:
\[ dG = V dP - S dT \]
Since the temperature change (dT) is zero in an isothermal process, the equation simplifies to:
\[ dG = V dP \]

### Step 4: Substitute Volume from Ideal Gas Law
Using the ideal gas law \( PV = nRT \), we can express the volume (V) as:
\[ V = \frac{nRT}{P} \]
Substituting this into the equation for dG gives:
\[ dG = \frac{nRT}{P} dP \]

### Step 5: Integrate to Find ΔG
To find the total change in Gibbs energy (ΔG), we need to integrate from the initial pressure to the final pressure:
\[ \Delta G = \int_{P_{initial}}^{P_{final}} \frac{nRT}{P} dP \]
Since \( nRT \) is constant during the process, we can take it out of the integral:
\[ \Delta G = nRT \int_{P_{initial}}^{P_{final}} \frac{1}{P} dP \]

### Step 6: Evaluate the Integral
The integral of \( \frac{1}{P} \) is:
\[ \int \frac{1}{P} dP = \ln P \]
Thus, we have:
\[ \Delta G = nRT \left( \ln P_{final} - \ln P_{initial} \right) \]
This can be simplified using the properties of logarithms:
\[ \Delta G = nRT \ln \left( \frac{P_{final}}{P_{initial}} \right) \]

### Step 7: Substitute Values
Now we can substitute the values:
- \( n = 1 \) mole (assuming we are calculating per mole)
- \( R = 8.314 \, J/(mol \cdot K) \)
- \( T = 293.15 \, K \)
- \( P_{final} = 2 \, bar \)
- \( P_{initial} = 1 \, bar \)

Calculating:
\[ \Delta G = 1 \times 8.314 \times 293.15 \times \ln \left( \frac{2}{1} \right) \]
\[ \Delta G = 8.314 \times 293.15 \times \ln(2) \]
Using \( \ln(2) \approx 0.693 \):
\[ \Delta G = 8.314 \times 293.15 \times 0.693 \]
\[ \Delta G \approx 8.314 \times 293.15 \times 0.693 \approx 1.7 \, kJ/mol \]

### Final Answer
The change in molar Gibbs energy of carbon dioxide gas at 20°C when it is isothermally compressed from 1.0 bar to 2.0 bar is approximately **1.7 kJ/mol**.

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