a fixed mass of gas at constant pressure occupies a volume.the gas undergoes a rise in temperature so that the Vrms of its molecule is doubled.What is the new volume
temperature is a measure of the average kinetic energy ... 1/2 m v^2
what happens to KE if v is doubled?
To find the new volume, we need to use the ideal gas law, which states:
PV = nRT
Where:
P is the pressure
V is the volume
n is the number of moles of gas
R is the ideal gas constant
T is the temperature
In this case, the pressure is constant, and the mass of the gas is fixed. Therefore, the number of moles of gas (n) remains the same.
Let's assume the initial temperature is T1, and the final temperature is T2. We are given that the root mean square velocity (Vrms) of the gas molecules has doubled.
The root mean square velocity (Vrms) of a gas is related to the temperature (T) by the equation:
Vrms ∝ sqrt(T)
Where "∝" denotes proportionality.
Since we are given that the Vrms is doubled, we have:
2Vrms = sqrt(T2) (Equation 1)
We can square both sides of the equation to solve for T2:
(2Vrms)^2 = T2
4(Vrms)^2 = T2 (Equation 2)
Now, let's simplify the equation PV = nRT and rewrite it in terms of T1 and V1 (initial temperature and volume) and T2 and V2 (final temperature and volume):
P × V1 = nR × T1 (Equation 3)
P × V2 = nR × T2 (Equation 4)
Since the pressure (P) and the number of moles (n) remain constant, we can divide Equation 4 by Equation 3 to eliminate the pressure and number of moles:
V2 / V1 = T2 / T1
Now, we can substitute the relationship between T2 and Vrms into the above equation:
V2 / V1 = (4(Vrms)^2) / T1 (from Equation 2)
Since we are given that the Vrms is doubled, we can substitute 2 times the initial Vrms for the final Vrms:
V2 / V1 = (4(2Vrms)^2) / T1
V2 / V1 = (4 × 4(Vrms)^2) / T1
V2 / V1 = 16(Vrms)^2 / T1
Now, we substitute the relationship between T1 and Vrms (from Equation 1):
V2 / V1 = 16(T1) / T1
V2 / V1 = 16
Therefore, the ratio of the final volume (V2) to the initial volume (V1) is 16. To find the new volume, you need to multiply the initial volume (V1) by 16:
V2 = 16 × V1
Thus, the new volume is 16 times the initial volume.