Q5.20P - Chemistry: Molecular Nature Of Matter And Change

Question

 What is the effect of the following on the volume of 1 mol of an ideal gas?

(a) The pressure is tripled (at constant T).

(b) The absolute temperature is increased by a factor of 3.0 (at constant P).

(c) Three more moles of the gas are added (at constant P and T).

Step-by-Step Solution

Answer

Answer

1. At constant temperature, if the pressure is tripled, then the new volume will be $\mathbf{1}\mathbf{/}\mathbf{3}$ of its initial volume.

2. At constant pressure, if the temperature is increased by the factor of 3.0, then the new volume will triple its initial volume.

3. At constant pressure and temperature, if three more moles of gas are added, then the new volume will be increased by a factor of 4.

Step 1: Subpart (a) When the pressure is tripled (at constant T)

The ideal gas law equation is,

$\mathit{P}\mathit{V}\mathbf{=}\mathit{n}\mathit{R}\mathit{T}$

Here,

P is pressure,

V is volume,

n is the number of moles,

R is the gas constant, and

T is the temperature.

At constant temperature, the pressure is tripled, then the new ideal gas equation is,

$\mathbf{3}\mathit{P}\mathit{V}\mathbf{\text{'}}\mathbf{=}\mathit{n}\mathit{R}\mathit{T}$

Now, divide the above equation by ideal gas law equation,

$$\begin{gathered} \frac{\mathbf{3}\mathbf{PV}\mathbf{\text{'}}}{\mathbf{PV}}\mathbf{=}\frac{\mathbf{nRT}}{\mathbf{nRT}} \\ \mathit{V}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathit{V} \end{gathered}$$

Thus, at constant temperature, if the pressure is tripled, then the new volume will be  of its initial volume.

Step 2: Subpart (b) If the absolute temperature is increased by a factor of 3.0 (at constant P)

The ideal gas law equation is,

$\mathbf{PV}\mathbf{=}\mathbf{nRT}$

At constant pressure, the temperature of the gas is increased by a factor of 3.0, then the new ideal gas equation is,

$\mathbf{PV}\mathbf{\text{'}}\mathbf{=}\mathbf{nR}\mathbf{3}\mathbf{T}$

Now, divide the above equation by the ideal gas law equation,

$$\begin{gathered} \frac{\mathbf{PV}\mathbf{\text{'}}}{\mathbf{PV}}\mathbf{=}\frac{\mathbf{nR}\mathbf{3}\mathbf{T}}{\mathbf{nRT}} \\ \mathit{V}\mathbf{\text{'}}\mathbf{=}\mathbf{3}\mathit{V} \end{gathered}$$

Thus, at constant pressure, if the temperature is increased by the factor of 3.0, then the new volume will triple its initial volume.

Step 3: Subpart (c) If three more moles of the gas are added (at constant P and T)

The ideal gas law equation is,

$\mathit{P}\mathit{V}\mathbf{=}\mathit{n}\mathit{R}\mathit{T}$

At constant pressure and temperature of the gas, three moles of gas are added, then the new ideal gas equation is,

$$\begin{gathered} \mathit{P}\mathit{V}\mathbf{\text{'}}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{3}\mathbf{)}\mathit{n}\mathit{R}\mathit{T} \\ \mathit{P}\mathit{V}\mathbf{\text{'}}\mathbf{=}\mathbf{4}\mathit{n}\mathit{R}\mathit{T} \end{gathered}$$

Now, divide the above equation by the ideal gas law equation,

$$\begin{gathered} \frac{\mathbf{PV}\mathbf{\text{'}}}{\mathbf{PV}}\mathbf{=}\frac{\mathbf{4}\mathbf{nRT}}{\mathbf{nRT}} \\ \mathit{V}\mathbf{\text{'}}\mathbf{=}\mathbf{4}\mathit{V} \end{gathered}$$

Thus, at constant pressure and temperature, three moles of gas are added, then the new volume will be increased by a factor of 4.