The volume of a monatomic ideal gas increases linearly with pressure, as shown in Fig. Calculate (a) increase in internal energy, (b) work done by the gas and (c ) heat supplied to the gas.

Explanation

We know that for a perfect gas, internal energy is given by
`U = (pV)/(gamma - 1)`
`implies Delta U = (p_(2)V_(2) - p_(1)V_(1))/(gamma - 1)`
`:. Delta U = (8 xx 10^(5) xx 0.5 - 4 xx 10^(5) xx 0.2)/(5//3 - 1) = 4.8 xx 10^(5) J`
Now, `Delta W = int_(0.2)^(0.5) PdV`
Here, p varies with V along a straight line.
Therefore, one way say2 `p = kV + V_(0)`
At `p = 4 xx 10^(5), V = 0.2 m^(2)` and `p = 8 xx 10^(5), V = 0.5 m^(3)`
`4 xx 10^(5) = k xx 0.2 + V_(0)` and `8 xx 10^(5) = k xx 0.5 + V_(0)`
Here, `k = (4)/(3) xx 10^(6)` and `V_(0) = (4)/(3) xx 10^(5)`
`p = (4)/(3) xx 10^(5) (10 V + 1)`
`Delta W = int_(0.2)^(0.5) (4)/(3) xx 10^(5) ((V^(2))/(2))_(0.2)^(0.5) + (4)/(3) xx 10^(5) (V)_(0.2)^(0.5)`
`(4)/(3) xx 10^(6) xx (0.21)/(2) + (4)/(3) xx 10^(5) xx 0.3 = 1.8 xx 10^(5) J`
By the first law of thermodynamics
`Delta Q = Delta U + Delta W`
`:. Delta Q = 4.8 10^(5) + 1.8 xx 10^(5) = 6.6 xx 10^(5) J`