Question

The densities of graphite and diamond at $298\, K$ are $2.25$ and $3.31\, g\, cm^{-3},$ respectively. If the standard free energy difference $(\Delta G^{\circ})$ is equal to $1895\, J\, mol^{-1}$, the pressure at which graphite will be transformed into diamond at $298\, K$ is -

• A. $9.92 \times 10^8 \, Pa $
• B. $9.92 \times 10^7 \, Pa $
• C. $9.92 \times 10^6 \, Pa $
• D. $9.92 \times 10^5 \, Pa $

Answer 1

The Correct Option is A

To determine the pressure at which graphite transforms into diamond at 298\, K, we use the relationship between Gibbs free energy change, pressure, and volume change.

The formula used is:

\Delta G = \Delta G^{\circ} + \Delta V \cdot \Delta P

Where:

• \Delta G is the Gibbs free energy change, which is zero at equilibrium.
• \Delta G^{\circ} = 1895 \, J \, mol^{-1} is the standard free energy difference.
• \Delta V is the change in volume when graphite turns into diamond.
• \Delta P is the pressure difference we need to calculate.

The change in volume, \Delta V, is given by:

\Delta V = \left(\frac{1}{\rho_{\text{diamond}}} - \frac{1}{\rho_{\text{graphite}}}\right) \cdot M

Where:

• \rho_{\text{diamond}} = 3.31 \, g \, cm^{-3}
• \rho_{\text{graphite}} = 2.25 \, g \, cm^{-3}
• M is the molar mass of carbon, approximately 12 \, g \, mol^{-1}.

First, calculate \Delta V:

\Delta V = \left(\frac{1}{3.31} - \frac{1}{2.25}\right) \cdot 12 \, cm^3 \, mol^{-1}

\Delta V = \left(0.302 - 0.444\right) \, \times 12 \, cm^3 \, mol^{-1}

\Delta V = -1.704 \, cm^3 \, mol^{-1}

We will convert this volume change from cm^3 to m^3:

\Delta V = -1.704 \, \times 10^{-6} \, m^3 \, mol^{-1}

Now substitute the values into the equation:

0 = 1895 \, J \, mol^{-1} + (-1.704 \, \times 10^{-6} \, m^3 \, mol^{-1}) \cdot \Delta P

Solving for \Delta P:

\Delta P = \frac{1895 \, J \, mol^{-1}}{1.704 \, \times 10^{-6} \, m^3 \, mol^{-1}}

\Delta P = 1.1119 \, \times 10^9 \, Pa

This value rounds to approximately 9.92 \times 10^8 \, Pa, matching the given correct option.

Hence, the correct answer is 9.92 \times 10^8 \, Pa.