Question

What is the effect of the following on the volume of \(1 \mathrm{~mol}\) of an ideal gas? (a) Temperature decreases from \(800 \mathrm{~K}\) to \(400 \mathrm{~K}\) (at constant \(P\) ). (b) Temperature increases from \(250^{\circ} \mathrm{C}\) to \(500^{\circ} \mathrm{C}\) (at constant \(P\) ). (c) Pressure increases from 2 atm to 6 atm (at constant \(T\) ).

Short answer

a) Volume is halved. b) Volume increases by about 1.48 times. c) Volume is reduced to one-third.

Step-by-step solution

- Understand the Ideal Gas Law

The Ideal Gas Law is given by: \[ PV = nRT \]where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

- Analyze Part (a)

For part (a), the temperature decreases from 800 K to 400 K at constant pressure (P). According to the Ideal Gas Law:\[ V = \frac{nRT}{P} \]Since \( n \), \( R \), and \( P \) are constant, the volume is directly proportional to the temperature. Hence:\[ V_1 / T_1 = V_2 / T_2 \]\[ V_2 = V_1 \times \frac{T_2}{T_1} = V_1 \times \frac{400}{800} = \frac{1}{2} V_1 \]This indicates that the volume is halved.

- Analyze Part (b)

For part (b), the temperature increases from 250 °C to 500 °C at constant pressure. Convert temperatures to Kelvin (K) first:\[ 250^{\circ} \mathrm{C} = 523 \mathrm{K} \] \[ 500^{\circ} \mathrm{C} = 773 \mathrm{K} \] Using the same direct proportionality principle:\[ V_2 = V_1 \times \frac{T_2}{T_1} = V_1 \times \frac{773}{523} \approx 1.48 V_1 \]The volume increases by a factor of approximately 1.48.

- Analyze Part (c)

For part (c), the pressure increases from 2 atm to 6 atm at constant temperature. According to the Ideal Gas Law:\[ V = \frac{nRT}{P} \]The volume is inversely proportional to the pressure, hence:\[ V_1 \times P_1 = V_2 \times P_2 \]\[ V_2 = V_1 \times \frac{P_1}{P_2} = V_1 \times \frac{2}{6} = \frac{1}{3} V_1 \]This indicates that the volume is reduced to one-third.

Key concepts

Temperature Effect on Gas Volume

The volume of an ideal gas is directly proportional to its temperature when pressure is held constant.
According to the Ideal Gas Law, when temperature decreases, the volume also decreases, and vice versa.
This is expressed mathematically as: \[ V = nRT / P \]
If the temperature is halved, the volume is halved as well.
By examining our example, a temperature drop from 800 K to 400 K results in the volume being halved.
Conversely, an increase from 250°C (523 K) to 500°C (773 K) increases the volume by a factor of 1.48, calculated as: \[ V_2 = V_1 \times \frac{T_2}{T_1} \approx 1.48 V_1. \] Ensuring we convert temperatures to Kelvin before use ensures accuracy.
Temperature changes have a straightforward, linear effect on gas volume when pressure remains unchanged.

Pressure Effect on Gas Volume

Pressure significantly impacts gas volume, but in an inverse manner.
Using the Ideal Gas Law, when temperature and the number of moles are constant, volume and pressure have an inverse relationship.
This means if the pressure is increased, the volume decreases, and if the pressure is decreased, the volume increases.
The relationship is given by: \[ V = \frac{nRT}{P} \] For example, if the pressure is tripled from 2 atm to 6 atm, the volume will reduce to one-third of its initial value: \[ V_2 = V_1 \times \frac{P_1}{P_2} = V_1 \times \frac{2}{6} = \frac{1}{3}V_1. \]
Understanding this inverse proportionality helps predict how a gas will behave under varying pressures.

Proportionality in Ideal Gas Law

The Ideal Gas Law, given by \[ PV = nRT \] describes the relationship between pressure (P), volume (V), number of moles (n), ideal gas constant (R), and temperature (T).
It highlights the proportionality among these variables.
Volume is directly proportional to temperature and inversely proportional to pressure.
Keep in mind:

• When temperature increases at constant pressure, volume increases.
• When pressure increases at constant temperature, volume decreases.

Both scenarios are clear cases of direct and inverse proportionality under controlled conditions.
Being familiar with these proportionality rules is crucial for predicting and understanding the behavior of ideal gases in different conditions.