A Sample of gas is held at constant pressure in a cylinder moved by a closed piston. If the volume is halved, how will the new Vrms speed compare with the old one?
a) square root of 2 times greater
b) the same
c) 2 times greater
d) 4 times greater
e) none
i think its e because i think it increases by square root of .5
but i really don't know.
Velocityrms = sqrt (3RT/M)
I don't see pressure anywhere in the formula. R is the same, T is the same(I assume), molar mass is the same. So Vrms must be ??
yes but T is affected volume and pressure changes according to the ideal gas law
The problem doesn't say T changes. Of course, it doesn't say it doesn't change, either. IF the volume was halved by moving the piston down then I could see that work would be done on the gas and T would increase. But the problem says the P stays constant and the volume is somehow (and miraculously) halved so I don't see how any work could be done on the gas. And no work, T should stay the same. The problem just isn't clear on that. You are right, however, the correct answer depends upon how T is treated.
To determine how the new root mean square (Vrms) speed of the gas will compare with the old speed when the volume is halved at constant pressure, we can use the ideal gas law and the principle of conservation of energy.
The ideal gas law relates the pressure, volume, and temperature of a gas:
PV = nRT
Where:
P = Pressure of the gas
V = Volume of the gas
n = Number of moles of gas
R = Ideal gas constant
T = Temperature of the gas
Since the problem states that the pressure is constant, we can simplify the ideal gas law as follows:
V ∝ T
When the volume is halved, it means that V' = V/2.
According to the principle of conservation of energy, the total energy of the gas remains constant.
The total energy of a gas is the sum of the kinetic energy of individual gas molecules.
The kinetic energy (Ek) of a gas molecule is given by:
Ek = (1/2)mv^2
Where:
m = Mass of the gas molecule
v = Velocity of the gas molecule
The root mean square (Vrms) speed is the square root of the average of the squares of the velocities of gas molecules.
Vrms = √(3RT/m)
Since the total energy of the gas remains constant, the average kinetic energy of the gas molecules also remains constant.
The average kinetic energy of the gas molecules is directly proportional to the temperature (T) of the gas, according to the kinetic theory of gases.
Therefore, when the volume is halved at constant pressure, the temperature of the gas remains constant.
Substituting the constant values, we have:
Vrms' = √(3RT'/m)
Vrms = √(3RT/m)
Since T and R are constant, we can compare the Vrms' and Vrms directly.
Vrms' = Vrms
Therefore, the new Vrms speed will be the same as the old speed when the volume is halved at constant pressure.
Therefore, the correct answer is b) the same.