Question

An ideal monatomic gas is contained in a vessel of constant volume 0.210 m3. The initial temperature and pressure of the gas are 300 K and 5.00 atm, respectively. The goal of this problem is to find the temperature and pressure of the gas after 13.0 kJ of thermal energy is supplied to the gas. (a) Use the ideal gas law and initial conditions to calculate the number of moles of gas in the vessel. mol (b) Find the specific heat of the gas. J/K (c) What is the work done by the gas during this process? kJ (d) Use the first law of thermodynamics to find the change in internal energy of the gas. kJ (e) Find the change in temperature of the gas. K (f) Calculate the final temperature of the gas. K (g) Use the ideal gas expression to find the final pressure of the gas. atm

Answer 1

(a) According to the ideal gas law –

The number of moles (n) of a gas is -

n = ( P1 ) ( V1 ) / ( R ) ( T1 )

n = ( 5.00 atm ) ( 210 L ) / ( 0.08206 atm - L / mol - K ) ( 300 K )

n = 42.65 moles

(b) Since the gas is monatomic.

So, Cpm = ( 5/2 ) ( R ) = ( 2.5 ) ( 8.314 J / mol - K ) = 20.785 J / mol - K

Cvm = ( 3/2 ) ( R ) = ( 1.5 ) ( 8.314 J /mol - K ) = 12.471 J / mol - K

(c) Since the volume is kept constant, therefore no work is done.

WB = 0.0 KJ

(d) First Law of Thermodynamics gives –

Q = Delta U + WB

Delta U = Q - WB = ( 13.0 kJ ) - ( 0.0 kJ ) = 13.0 kJ

(e) Delta U = ( n ) ( Cvm ) ( T2 - T1 )

T2 - T1 = ( Delta U ) / ( n ) ( Cvm )

T2 - T1 = ( 13000 J ) / ( 42.65 mol ) ( 12.471 J / mol - K )

T2 - T1 = 24.4 K

(f) T2 = T1 + ( T2 - T1 ) = 300 K + 24.4 K = 324.4 K

(g) P2 = ( n ) ( R ) ( T2 ) / ( V2 )

=> P2 = ( 42.65 mol ) ( 0.08206 atm - L / mol - k ) ( 324.4 K ) / ( 210 L )

P2 = 5.406 atm