A monatomic ideal gas is initially at a pressure of 180 atm

A monatomic ideal gas is initially at a pressure of 1.80 atm in a 1.90-L cylindrical container with a piston on one side. It is compressed to a volume of 0.300 L. (a) Plot the pressure versus volume during this process, assuming the process is either (i) adiabatic and (ii) isothermal.

(b) Determine the final pressure, assuming the process is either (i) adiabatic or (ii) isothermal.

adiabatic 4e6	Pa
isothermal ?	Pa

Solution

Solution:-

A) Adiabatic Process

We know equation for the ideal gas is:

PV=nRT (1)

adiabatic process is the one in which no heat exchange with the environment

U= Q+W (2)

which means Q=0, Putting this in equation 2 we get

U=W

From equation (1) we know that for the adiabaic process is :

PV^5/2= Constant (C) (3)

To calculate the constant we have the values for the initial conditions:

(182385)*(0.0019)^5/3 = C

***Values are written after converting atm to pascals and L to m^3

After solving further we get the values as C=5.3 which will be used for future reference

Now we know the value for the constant during the final stage we have the values for the volume and the constant, we can easily calculte the value for pressure at final stage

V(initial) = 182385Pa

P(initial) = 0.0019m^3

Similarly, V(final) = .0003m^3

P(final) = X

To calculte the value for X we will now use the equation (3)

P = C/V^(5/2)

Putting the values in this equation we get:-

X= 5.3/(0.0003)^(5/3) = 3955223.88 Pa = 39.035 atm

B) Isothermal Process

To calculate the Isothermal process we know the equation is

PV=Constant (C) (4)

Similar to as done above we can calulate using equation (4):-

(182385)*(0.0019) = C

C=346.53

V(initial) = 182385Pa

P(initial) = 0.0019m^3

Similarly, V(final) = .0003m^3

P(final) = X

To calculte the value for X we will now use the equation (4)

P = C/V

Putting the values in this equation we get:-

X= 346.53/(0.0003) = 1155100 Pa = 11.39 atm

Now we have the values for intitial and final conditions for the Pressure and the volume and we can plot accordingly

** one can use online tools for the ploting as values have been drieved here.

Solution B)

Using the similar Equations as given above in the previous solution:-

we can find out the values for the final pressure:-

Adiabatic:-

PV^5/2=C

(4*10^6)*(0.0003)^5/3=C

C=5.377

Now, V(final)=0.0001, P=(5.377)/(0.0001)^5/3

P=24962859.8

Isothermal:-

PV=C

(1*10^6)*(0.0003) =C

C=300

Now, V(final)=0.0001, P=(300)/(0.0001)

C=3000000