Q69PE - College Physics (Urone)

Question

Question: (a) How many moles per cubic meter of an ideal gas are there at a pressure of 1.00×10¹⁴ N/m² and at 0ºC? (b) What is unreasonable about this result? (c) Which premise or assumption is responsible?

Step-by-Step Solution

Answer

Answer

1. The number of moles per cubic meter is  $4.40\times {10}^{10}mol/m^3$
2. The result is unreasonable because it is a higher value.
3. This gas can’t be considered an ideal one.

Step 1: Concept Introduction

The ideal gas law relation between pressure, volume, and temperature of an ideal gas. This is represented as;

$$\begin{gathered} \mathit{P}\mathit{V}\mathbf{=}\mathit{n}\mathit{R}\mathit{T} \\ \frac{\mathbf{n}}{\mathbf{V}}\mathbf{=}\frac{\mathbf{P}}{\mathbf{R}\mathbf{T}} \end{gathered}$$

Here, is the pressure, is the volume, is the number of moles, is the universal gas constant and is the temperature. Pressure is given as 1.00×10¹⁴ N/m². The temperature is given as 0°C. This is equal to a temperature of 273.15 K. The value of is 8.314 JK^-1mol^-1.

Step 1: Solution for part (a)

Substitute these values in the equation.

$$\begin{gathered} \frac{n}{v}=\frac{1.00\times {10}^{14}N/m^2}{8.314J{K}^{-1}mo{l}^{-1}\times 273.15K} \\ =4.40\times {10}^{10}mol/m^3 \end{gathered}$$

Therefore, the number of moles per cubic meter is  $4.40\times {10}^{10}mol/m^3$

Step 2: Solution for part (b)

The result is unreasonable. Because the number of moles is the ratio of mass to molecular mass, and according to this result, the density of this ideal gas will be that much higher. There is no such gas that exists that has this much density.

Step 3: Solution for part (c)

The concept of an ideal gas is not valid here. Higher pressure is the reason for the high value of the mole per cubic meter. If the pressure was lower than this, the result may have been reasonable.