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Fundamentals Of Physics
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Short Answer

An ideal monatomic gas initially has a temperature of 330K and a pressure of 6.00atm. It is to expand from volume 500cm3 to volume1500cm3. If the expansion is isothermal, what are (a) the final pressure and (b) the work done by the gas? If, instead, the expansion is adiabatic, what are (c) the final pressure and (d) the work done by the gas?

  1. Final Pressure when expansion process is isothermal is 2.00atm.
  2. Work done by the gas when expansion process is isothermal is 333J.
  3. Final Pressure when expansion process is adiabatic is 0.96atm.
  4. Work done by the gas when expansion process is adiabatic is 236J.
See the step by step solution

Step by Step Solution

Step 1: Write the given data from the question:

Initial Temperature;Ti=330K

Initial Pressure;Pi=6.00atm

Initial Volume;Vi=500cm3

Final Volume; Vf=1500cm3

Step 2: Understanding the concept

In case of isothermal process temperature remains constant. The expression for the work done in case of isothermal process is given by,

W=nRTlnvfvi…… (i)

Here W is the work done, n is the number of moles, R is the gas constant, T is the temperature, vf is the final volume of the gas, vi is the initial volume of the gas.

From the ideal gas equation;

PV=nRT.........(ii)

Here P is the pressure and V is the volume.

From equation (i) and (ii)

W=PiVilnvfvi

Step 3: (a) Calculate the final pressure when the expansion is isothermal

The process is isothermal, we can say that,PiVi=PfVf

PiVi=PfVf

Substitute 6.00atm for Pi,500cm3 for Vi, 1500cm3 for Vf into the above equation,

(6.00×500)=Pf×1500Pf=(6.00×500)1500

Pf=2.00atm

Therefore the final pressure when the expansion is isothermal is 2.00atm.

Step 4: (b) Calculate the work done by the gas when the expansion is isothermal

The process is isothermal, we can say that

W=PiVilnvfvi

We have to convert the pressure to Pascal from and Volume to m3 from cm3

Initial pressure

Pi=6atm=6atm×101×105pa1atm=6.06×105

Initial volume

vi=500cm3=500cm3×1×10-6m31cm3=5×10-4m3

Final volume

vt=1500cm3=1500cm3×1×10-6m31cm3=15×10-4m3

The expression for the work done in case of isothermal process is given by,

Substitute 6.06×105pa for Pi,5×10-4m3 for Vi,15×10-4m3for Vf into the above equation,

w=6.06x105×5×10-4×In15×10-45×10-4=332.87=333J

Therefore the work done by the gas when the expansion is isothermal is 333J.

Step 5: (c) Calculate the final pressure when the expansion is adiabatic

The process is adiabatic so, we can say that,

PiViγ=PfVfγ

(6.00×5001.67)=Pf×15001.67

Substitute 6.00 atm for Pi,500cm3 for Vi,1500cm3 for Vf,1.67 for γ into the above equation,

Pf=(6.00×5001.67)15001.67=0.957=0.96atm

The final pressure when the expansion is adiabatic is 0.96atm.

Step 6: (d) Calculate the work done by the gas when the expansion is adiabatic

The process is adiabatic so, we can say that,

W=PiViγViVfV-γdV

On integrating the above equation,

localid="1662538514457" w=PiVi×Vf1-γ1-γ=PfVf-PiVi1-γ

Convert the initial and final pressure from atm to pa

Initial pressure

localid="1662539084992" Pi=6atm=6atm×101325pa1atm=607950paPf=0.96atm=0.96atm×101325pa1atm=97272pa

Final pressure

Pf=0.96atm=0.96atm×101325pa1atm=97272pa

The expression for work done by the gas in case of adiabatic process is calculated above;

W=(PfVf-PiVi)/(1-γ)

Substitute 97272pa for Pf,607950pa for Pi,5×10-4m3 for Vi and 15×10-4m3forVf into the above equation,

w=97272×15×10-4-607950×5×10-41-1.67=235.9=236J

Therefore the work done by the gas when the expansion is adiabatic is 236J.

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