Question

A rigid container holds 4.00 mol of a monatomic ideal gas that has temperature 300 K. The initial pressure of the gas is 6.00 * 104 Pa. What is the pressure after 6000 J of heat energy is added to the gas?

Answer 1

Answer:

The final pressure of the monoatomic ideal gas is 8.406 × 10⁶ pascals.

Explanation:

When a container is rigid, the process is supposed to be isochoric, that is, at constant volume. Then, the equation of state for ideal gases can be simplified into the following expression:

$\frac{P_{1}}{T_{1}} = \frac{P_{2}}{T_{2}}$

Where:

$P_{1}$, $P_{2}$ - Initial and final pressures, measured in pascals.

$T_{1}$, $T_{2}$ - Initial and final temperatures, measured in Kelvins.

In addtion, the specific heat at constant volume for monoatomic ideal gases, measured in joules per mole-Kelvin is given by:

$\bar c_{v} = \frac{3}{2}\cdot R_{u}$

Where:

$R_{u}$ - Ideal gas constant, measured by pascal-cubic meters per mole-Kelvin.

If $R_{u} = 8.314\,\frac{Pa\cdot m^{3}}{mol\cdot K}$, then:

$\bar c_{v} = \frac{3}{2}\cdot \left(8.314\,\frac{Pa\cdot m^{2}}{mol\cdot K} \right)$

$\bar c_{v} = 12.471\,\frac{J}{mol\cdot K}$

And change in heat energy ($Q$), measured by joules, by:

$Q = n\cdot \bar c_{v}\cdot (T_{2}-T_{1})$

Where:

$n$ - Molar quantity, measured in moles.

The final temperature of the monoatomic ideal gas is now cleared:

$T_{2} = T_{1} + \frac{Q}{n\cdot \bar c_{v}}$

Given that $T_{1} = 300\,K$, $Q = 6000\,J$, $n = 4\,mol$ and $\bar c_{v} = 12.471\,\frac{J}{mol\cdot K}$, the final temperature is:

$T_{2} = 300\,K + \frac{6000\,J}{(4\,mol)\cdot \left(12.471\,\frac{J}{mol\cdot K} \right)}$

$T_{2} = 420.279\,K$

The final pressure of the system is calculated by the following relationship:

$P_{2} = \left(\frac{T_{2}}{T_{1}}\right) \cdot P_{1}$

If $T_{1} = 300\,K$, $T_{2} = 420.279\,K$ and $P_{1} = 6.00\times 10^{4}\,Pa$, the final pressure is:

$P_{2} = \left(\frac{420.279\,K}{300\,K} \right)\cdot (6.00\times 10^{4}\,Pa)$

$P_{2} = 8.406\times 10^{4}\,Pa$

The final pressure of the monoatomic ideal gas is 8.406 × 10⁶ pascals.