How Does Increasing RMS Speed Affect Gas Temperature?

[Nghi]

Homework Statement

The rms speed of a sample of gas is increased by 8 percent.

(a) By how much does the gas's temperature change? (given in %)
(b) By how much does the gas's temperature change, assuming its volume is held constant? (%)

Homework Equations

rms = $$\sqrt{3KT/M}$$

The Attempt at a Solution

I have no idea how to solve this. I set the left side of the equation equal to 0.08, and then I got lost from there. -__-

 

[anathema]

Thank you for your question. Let me guide you through the solution step by step.

(a) To find the change in temperature, we first need to understand how the rms speed is related to temperature. The rms speed is directly proportional to the square root of the temperature, as seen in the equation rms = \sqrt{3KT/M}. This means that if we increase the rms speed by 8 percent, the temperature will also increase by 8 percent. This can be written as:

\Delta T = 0.08T

(b) Now, let's assume that the volume is held constant. In this case, the ideal gas law can be used, which states that PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is the temperature. Since the volume is constant, we can rearrange the equation to get:

T = \frac{PV}{nR}

Now, let's substitute this into the equation for rms speed:

rms = \sqrt{3K\frac{PV}{nR}\frac{1}{M}}

To find the change in temperature, we need to find the new rms speed and then solve for T. Since the rms speed has increased by 8 percent, the new rms speed will be 1.08 times the original rms speed.

rms_{new} = 1.08rms_{original}

Substituting this into the equation for rms speed and solving for T, we get:

T_{new} = 1.08^2T_{original}

This means that the temperature will increase by 8.64 percent. We can write this as:

\Delta T = 0.0864T

I hope this helps you understand the solution better. Please let me know if you have any further questions. Good luck with your studies!

 

[ogb p]

As a scientist, let's break down the problem and approach it step by step.

First, let's define some terms. rms speed refers to the root-mean-square speed, which is a measure of the average speed of particles in a gas. It is calculated using the equation rms = √(3KT/M), where K is the Boltzmann constant, T is the temperature in Kelvin, and M is the molar mass of the gas.

Now, let's address part (a) of the problem. We are given that the rms speed of the gas has increased by 8%, so we can write this as:

rms_new = 1.08 * rms_old

Since the temperature (T) is directly proportional to the rms speed (rms), we can say that the temperature has also increased by 8%. Therefore, the gas's temperature change is 8%.

For part (b), we need to assume that the volume of the gas is held constant. This means that the number of particles (n) and the molar mass (M) are also constant. Using the ideal gas law, PV = nRT, we can rearrange it to get T = PV/nR.

Since the volume (V) and the number of particles (n) are constant, the temperature (T) is directly proportional to the pressure (P). Therefore, if the rms speed increases by 8%, the temperature will also increase by 8%.

In conclusion, the gas's temperature change is 8% for both parts (a) and (b) of the problem.