Question

A well insulated rigid container of volume 1 m³ contains 1.0 kg of an ideal gas [C_p = 1000 J/(kg.K) and C_v = 800 J/(kg.K)] at a pressure of \({10^5}\)Pa. A stirrer is rotated at constant rps in the container for 1000 rotations and the applied torque is 100 N-m. The final temperature of the gas (in K) is _________.

Answer 1

Work done = T.ω 

= 100 × 2π × 1000 

\(\Rightarrow {\rm}W.D. = \frac{{100 \times 2\pi \times 1000}}{{1000}} = 628.3185\ kJ\)

And P₁V₁ = mRT₁

\(\Rightarrow {10^5} \times 1 = 1 \times \left( {1000 - 800} \right){T_1} \Rightarrow {T_1} = 500K\)

Now from the first law, dQ = dU + dW

since, dQ = 0 and dU = C_vdT and work done is on the system i.e. dW  = - Ve

So, W.D. = C_vdT ⇒ \(dT = \frac{628.3185}{0.8}= 785 \ K\)

Then \({T_2} = 500 + 785 = 1285\ K\)