Chapter 2: Q72P (page 542)

A long, closed cylindrical tank contains a diatomic gas that is maintained at a uniform temperature that can be varied. When you measure the speed of sound v in the gas as a function of the temperature T of the gas, you obtain these results:

(a) Explain how you can plot these results so that the graph will be well fit by a straight line. Construct this graph and verify that the plotted points do lie close to a straight line. (b) Because the gas is diatomic, g = 1.40. Use the slope of the line in part (a) to calculate M, the molar mass of the gas. Express M in grams/mole. What type of gas is in the tank?

Short Answer

Expert verified

A) The equation that states the straight line of the graph is $v^{2} = (\sqrt{\frac{γR}{M}}) T$

B) $M = 46.5 g / mol$

Step by step solution

Relation between the speed of sound v and the temperature T

The speed of sound v and the temperature T is $v = \sqrt{\frac{γ R T}{M}}$ where $γ$ is the ratio of heat capacities, M is the molar mass and R is gas constant

Graph between v and T

Take the square of both sides of the above equation, so we can plot a graph between v and T

$v^{2} = (\sqrt{\frac{γR}{M}}) T$

Plot a graph between $v^{2}$ and T. As shown in the figure below, the graph is a straight line.

Calculate the slope of the graph

The slope of the graph is $(\sqrt{\frac{γR}{M}})$ For the diatomic gas $γ = 1$ and $R = 8.314 J / mol . k$ Now, we find the slope from the curve and solve the equation for M

$Slope = \frac{14 * 10^{4} m^{2} / s^{2} - 13.5 m^{2} / s}{340 K - 320 K} = 250 m^{2} / {Ks}^{2} M = (\frac{1.40 \times (8.3114 J / molK)}{250 m^{2} / {Ks}^{2}}) = 0.0465 Kg / mol = 46.5 g / mol$