Question

An ideal gas will have maximum density when ___________.

• A. $\mathrm{P}=1\mathrm{atm},\mathrm{T}=300\mathrm{K}$
• B. $\mathrm{P}=2\mathrm{atm},\mathrm{T}=150\mathrm{K}$
• C. $\mathrm{P}=0.5\mathrm{atm},\mathrm{T}=600\mathrm{K}$
• D. $\mathrm{P}=1\mathrm{atm},\mathrm{T}=500\mathrm{K}$

Answer 1

The correct option is B

$\mathrm{P}=2\mathrm{atm},\mathrm{T}=150\mathrm{K}$

Explanation for the correct option

Option B: $\mathrm{P}=2\mathrm{atm},\mathrm{T}=150\mathrm{K}$

Step 1: Given data

An ideal gas.

We need to find when the ideal gas will have maximum density.

Step 2: Formula used

The ideal gas equation: $\mathrm{PV}=\mathrm{nRT}$, where

$\mathrm{P}=\mathrm{Pressure}$

$\mathrm{V}=\mathrm{Volume}$

$\mathrm{n}=\mathrm{Number}\mathrm{of}\mathrm{moles}=\frac{\mathrm{m}(\mathrm{given}\mathrm{mass})}{\mathrm{M}(\mathrm{Molar}\mathrm{mass})}$

$\mathrm{R}=\mathrm{Gas}\mathrm{constant}$

$\mathrm{T}=\mathrm{Temperature}$

Density is calculated using the formula: $\mathrm{\rho }=\frac{\mathrm{m}}{\mathrm{v}}$ where

$\mathrm{\rho }=\mathrm{density}$

$\mathrm{m}=\mathrm{mass}\mathrm{of}\mathrm{the}\mathrm{substance}$

$\mathrm{V}=\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{substance}$

Step 3: Calculation of densities

Using the ideal gas equation:

$$\begin{gathered} \Rightarrow \mathrm{PV}=\frac{\mathrm{m}}{\mathrm{M}}\mathrm{RT} \\ \Rightarrow \frac{\mathrm{PM}}{\mathrm{RT}}=\frac{\mathrm{m}}{\mathrm{V}} \end{gathered}$$

The formula for density is $\mathrm{\rho }=\frac{\mathrm{m}}{\mathrm{v}}$

$\Rightarrow \mathrm{\rho }=\frac{\mathrm{PM}}{RT}$

$\Rightarrow \mathrm{\rho }\propto \frac{\mathrm{P}}{T}$

Therefore for maximum density, the pressure to temperature ratio should be maximum.

$\Rightarrow \frac{\mathrm{P}}{T}=\frac{2}{150}=0.0133$

Hence, the pressure to temperature ratio is maximum for option B. Therefore it is the correct option.

Explanation for incorrect options

Option A: $\mathrm{P}=1\mathrm{atm},\mathrm{T}=300\mathrm{K}$

Step 1: Given data

An ideal gas.

We need to find when the ideal gas will have maximum density.

Step 2: Formula used

The ideal gas equation: $\mathrm{PV}=\mathrm{nRT}$, where

$\mathrm{P}=\mathrm{Pressure}$

$\mathrm{V}=\mathrm{Volume}$

$\mathrm{n}=\mathrm{Number}\mathrm{of}\mathrm{moles}=\frac{\mathrm{m}(\mathrm{given}\mathrm{mass})}{\mathrm{M}(\mathrm{Molar}\mathrm{mass})}$

$\mathrm{R}=\mathrm{Gas}\mathrm{constant}$

$\mathrm{T}=\mathrm{Temperature}$

Density is calculated using the formula: $\mathrm{\rho }=\frac{\mathrm{m}}{\mathrm{v}}$ where

$\mathrm{\rho }=\mathrm{density}$

$\mathrm{m}=\mathrm{mass}\mathrm{of}\mathrm{the}\mathrm{substance}$

$\mathrm{V}=\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{substance}$

Step 3: Calculation of densities

Using the ideal gas equation:

$$\begin{gathered} \Rightarrow \mathrm{PV}=\frac{\mathrm{m}}{\mathrm{M}}\mathrm{RT} \\ \Rightarrow \frac{\mathrm{PM}}{\mathrm{RT}}=\frac{\mathrm{m}}{\mathrm{V}} \end{gathered}$$

The formula for density is $\mathrm{\rho }=\frac{\mathrm{m}}{\mathrm{v}}$

$\Rightarrow \mathrm{\rho }=\frac{\mathrm{PM}}{RT}$

$\Rightarrow \mathrm{\rho }\propto \frac{\mathrm{P}}{T}$

Therefore for maximum density, the pressure to temperature ratio should be maximum.

$\Rightarrow \frac{\mathrm{P}}{T}=\frac{1}{300}=0.0033$

Hence, the pressure to temperature ratio is not maximum for option A. Therefore it is not the correct option.

Option C: $\mathrm{P}=0.5\mathrm{atm},\mathrm{T}=600\mathrm{K}$

Step 1: Given data

An ideal gas.

We need to find when the ideal gas will have maximum density.

Step 2: Formula used

The ideal gas equation: $\mathrm{PV}=\mathrm{nRT}$, where

$\mathrm{P}=\mathrm{Pressure}$

$\mathrm{V}=\mathrm{Volume}$

$\mathrm{n}=\mathrm{Number}\mathrm{of}\mathrm{moles}=\frac{\mathrm{m}(\mathrm{given}\mathrm{mass})}{\mathrm{M}(\mathrm{Molar}\mathrm{mass})}$

$\mathrm{R}=\mathrm{Gas}\mathrm{constant}$

$\mathrm{T}=\mathrm{Temperature}$

Density is calculated using the formula: $\mathrm{\rho }=\frac{\mathrm{m}}{\mathrm{v}}$ where

$\mathrm{\rho }=\mathrm{density}$

$\mathrm{m}=\mathrm{mass}\mathrm{of}\mathrm{the}\mathrm{substance}$

$\mathrm{V}=\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{substance}$

Step 3: Calculation of densities

Using the ideal gas equation:

$$\begin{gathered} \Rightarrow \mathrm{PV}=\frac{\mathrm{m}}{\mathrm{M}}\mathrm{RT} \\ \Rightarrow \frac{\mathrm{PM}}{\mathrm{RT}}=\frac{\mathrm{m}}{\mathrm{V}} \end{gathered}$$

The formula for density is $\mathrm{\rho }=\frac{\mathrm{m}}{\mathrm{v}}$

$\Rightarrow \mathrm{\rho }=\frac{\mathrm{PM}}{RT}$

$\Rightarrow \mathrm{\rho }\propto \frac{\mathrm{P}}{T}$

Therefore for maximum density, the pressure to temperature ratio should be maximum.

$\Rightarrow \frac{\mathrm{P}}{T}=\frac{0.5}{600}=0.00083$

Hence, the pressure to temperature ratio is not maximum for option C. Therefore it is not the correct option.

Option D: $\mathrm{P}=1\mathrm{atm},\mathrm{T}=500\mathrm{K}$

Step 1: Given data

An ideal gas.

We need to find when the ideal gas will have maximum density.

Step 2: Formula used

The ideal gas equation: $\mathrm{PV}=\mathrm{nRT}$, where

$\mathrm{P}=\mathrm{Pressure}$

$\mathrm{V}=\mathrm{Volume}$

$\mathrm{n}=\mathrm{Number}\mathrm{of}\mathrm{moles}=\frac{\mathrm{m}(\mathrm{given}\mathrm{mass})}{\mathrm{M}(\mathrm{Molar}\mathrm{mass})}$

$\mathrm{R}=\mathrm{Gas}\mathrm{constant}$

$\mathrm{T}=\mathrm{Temperature}$

Density is calculated using the formula: $\mathrm{\rho }=\frac{\mathrm{m}}{\mathrm{v}}$ where

$\mathrm{\rho }=\mathrm{density}$

$\mathrm{m}=\mathrm{mass}\mathrm{of}\mathrm{the}\mathrm{substance}$

$\mathrm{V}=\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{substance}$

Step 3: Calculation of densities

Using the ideal gas equation:

$$\begin{gathered} \Rightarrow \mathrm{PV}=\frac{\mathrm{m}}{\mathrm{M}}\mathrm{RT} \\ \Rightarrow \frac{\mathrm{PM}}{\mathrm{RT}}=\frac{\mathrm{m}}{\mathrm{V}} \end{gathered}$$

The formula for density is $\mathrm{\rho }=\frac{\mathrm{m}}{\mathrm{v}}$

$\Rightarrow \mathrm{\rho }=\frac{\mathrm{PM}}{RT}$

$\Rightarrow \mathrm{\rho }\propto \frac{\mathrm{P}}{T}$

Therefore for maximum density, the pressure to temperature ratio should be maximum.

$\Rightarrow \frac{\mathrm{P}}{T}=\frac{1}{500}=0.002$

Hence, the pressure to temperature ratio is not maximum for option D. Therefore it is not the correct option.

Hence option B, $\mathrm{P}=2\mathrm{atm},\mathrm{T}=150\mathrm{K}$ is the correct answer.