A 10.0 cm diameter cylinder contains 0.180 motet of Nitrogen (N_2) gas. A piston
ID: 3163121 • Letter: A
Question
A 10.0 cm diameter cylinder contains 0.180 motet of Nitrogen (N_2) gas. A piston can slide in and out of the cylinder, the other end is scaled. The cylinder's initial length is 0 500 m at an initial pressure of 1.00 artist. The gas is adiabatically compressed until the final length of the cylinder is 0.300 m. The speed of sound in the gas at the initial temperature and pressure is 335 m/s. The effective radius of the nitrogen molecules is 0.15 nm (a) Calculate the initial temperature of the nitrogen gas, in Celsius. Calculate the pressure of the nitrogen gas at the end of this process (the "initial" state) Calculate the mean free path of the nitrogen molecules at the of this Calculate the frequency of the third vibrational mode for sound in the piston, at the beginning of this process (the "initial" state). (the "initial" state).Explanation / Answer
Given data
diameter of cylinder, 2r = 0.1 m
number of moles of nitrogen gas, n = 0.18 moles
initial length of cylinder, lo = 0.5 m
final length, l = 0.3 m
initial pressure, Po = 1 atm
speed of sound in intial conditions, v = 335 m/s
initial volume of gas = A*lo [ A is cross section area of the cylinder]
NOw speed of sound , v = sqroot(gamma*RT/M)
T is temperature, R is gas constant, M is molar mass, gamma is specific heat ratio
for N2, gamma = 1 + 2/f = 1 + 2/5 = 1.4
M = 0.014 kg/mol
so, 335 = sqroot(1.4*8.31*T/0.014)
a) T = 135.048 K = -138.11 C[ initial temperature]
b) At the end of the adiabatic process, pressure = P'
for adiabatic process, PV^gamma = constant
Po*Vo^(gamma) = P'*V'^(Gamma)
P' = Po(Alo/Al)^gamma = Po(0.5/0.3)^1.4 = 2.044 Po = 2.044 atm
c) Mean free path of a molecule, lambda = kb*T/sqroot(2)*pi*d^2*p
Kb = boltzmann constant = 1.38*10^-23
T = temperature (absolute)
d = diameter of nitrogen molecule = 0.3*10^-9 m
p = pressure = 1 atm = 1.01*10^5 Pa
at the start of process
lambda = 4.6169*10^-5 mm
d) aT THE begining of the process
lo = 0.5 m
third vibrational mode of sound, i.e.
lambda = 2l/3 = 2*0.5/3 = 0.33 m
v = 335 m/s
so, frequency, f = ?
v = lambda*f
f = v/lambda = 1015.15 Hz
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.