What Is the New Volume When RMS Velocity Doubles?

[Amith2006]

Sir,
Please help me with this problem.
# A fixed mass of gas at constant pressure occupies a volume V. The gas undergoes a rise in temperature so that the root mean squared velocity(c) of the molecule is doubled. What is the new volume?
I solved it in the following way:-
Let V1 be the initial volume & V2 be the final volume. Assuming the pressure to be constant,
c = (3PV1/M)^1/2 --------- (1)
2c = (3PV2/M)^1/2 --------- (2)
Dividing equation (1) by (2) we get,
½ = (V1/V2)^1/2
Squaring on both sides we get,
¼ = (V1/V2)
V2 = 4V1
But the answer given in my book is V2 = V1/[(2)^1/2] read as V1 divided by root 2.Here the symbol ^ represents power.

 

[Doc Al]

Your answer seems OK to me. Since rms speed is proportional to the square root of the temperature, the temperature (and volume) must increase by a factor of four.

The book's answer makes no sense, since it implies that the volume decreases.

 

[Andrew Mason]

Sir,
Please help me with this problem.
# A fixed mass of gas at constant pressure occupies a volume V. The gas undergoes a rise in temperature so that the root mean squared velocity(c) of the molecule is doubled. What is the new volume?
I solved it in the following way:-
Let V1 be the initial volume & V2 be the final volume. Assuming the pressure to be constant,
c = (3PV1/M)^1/2 --------- (1)
2c = (3PV2/M)^1/2 --------- (2)
Dividing equation (1) by (2) we get,
½ = (V1/V2)^1/2
Squaring on both sides we get,
¼ = (V1/V2)
V2 = 4V1
But the answer given in my book is V2 = V1/[(2)^1/2] read as V1 divided by root 2.Here the symbol ^ represents power.

Where did you get your equations (1) and (2)?

How is temperature related to the v_rms of the gas? Temperature is a measure of the internal (kinetic) energy of the gas molecules. So how is the kinetic energy of the molecules related to speed?

If v_rms of the gas doubles, what is the increase in temperature (internal kinetic energy)?

Use PV=nRT with P constant to determine how Volume changes with temperature.

AM

 

[Amith2006]

Sir,
Please help me with this problem.
# A fixed mass of gas at constant pressure occupies a volume V. The gas undergoes a rise in temperature so that the root mean squared velocity(c) of the molecule is doubled. What is the new volume?
I solved it in the following way:-
Let V1 be the initial volume & V2 be the final volume. Assuming the pressure to be constant,
c = (3PV1/M)^1/2 --------- (1)
2c = (3PV2/M)^1/2 --------- (2)
Dividing equation (1) by (2) we get,
½ = (V1/V2)^1/2
Squaring on both sides we get,
¼ = (V1/V2)
V2 = 4V1
But the answer given in my book is V2 = V1/[(2)^1/2] read as V1 divided by root 2.Here the symbol ^ represents power.

Thank you Sir for your help.