Question

Please Help! Let P and V represent the pressure and volume of Xe(g). If a piston is used to reduce the volume of the gas to V/2 at a constant temperature, what is the new pressure in the container in terms of the original pressure, P?
What happens to the average speed of the Xe (g) atoms as the original volume is reduced to V/2 at a constant temperature? Explain.

Answer 1

1) 2P

2) Average speed does not change

Explanation:

1)

To solve the first part of the problem, we can use Boyle's Law, which states that:

"For a fixed mass of an ideal gas kept at constant temperature, the pressure of the gas is inversely proportional to its volume"

Mathematically:

$pV=const.$

where

p is the pressure of the gas

V is its volume

For the Xe gas in this problem we can write

$p_1 V_1 = p_2 V_2$

where:

$p_1 = P$ is the initial pressure

$V_1=V$ is the initial volume

$V_2=\frac{V}{2}$ is the final volume

Solving for p2, we find the final pressure:

$p_2=\frac{P V_1}{V_2}=\frac{pV}{V/2}=2P$

So, the pressure has doubled.

2)

The average speed of the atoms/molecules in the gas is given by the formula

$v_{rms}=\sqrt{\frac{3RT}{M}}$

where

R is the gas constant

T is the absolute temperature (in Kelvin) of the gas

M is the molar mass of the gas

$v_{rms}$ is known as rms speed of the particles in the gas

From the formula, we see that the speed of the atoms in the gas depends only on the temperature of the gas.

In the Xe(g) gas in this problem, the temperature is kept constant; therefore, since nothing changes in the formula, this means that the average speed also does not change.

Answer 2

The pressure of the gas is doubled and the average speed of the gas molecules does not change.

According to Boyle's law, the volume of a given mass of gas is inversely proportional to the pressure at constant temperature. Now;

Initial pressure of the gas (P1) = P

Initial volume of the gas(V1) = V

Final volume of the gas (V2) = V/2

Final pressure of the gas (P2) = ?

Using the relation of Boyle's law;

P1V1 = P2V2

P2 =  P1V1/V2

P2 = P × V/V/2

P2 = 2P

The pressure is doubled

Regarding the average speed of gas molecules, the average speed of a gas molecules depends on the temperature from the relation;

$vrms = \sqrt{\frac{3RT}{M} }$

If the temperature is held constant, the average speed of gas molecules is does not change.

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