Question

When 100 J of heat is given to an ideal gas it expands from $200c{m}^3$ to $400c{m}^3$ at a constant pressure of $3\times {10}^3$ Pa.Calculate (a) the change in internal energy of the gas, (b) the number of moles in the gas if the initial temperature is 400 k, (c) the molar heat capacity $C_p$ at constant pressure and (d) the molar heat capacity $C_r$ at constant volume .$[R=\frac{25}{3}J/mol-k]$

Answer 1

$Q=+100$

By $1^{st}law$

(a) $\Delta Q=\Delta E+dw$

$100=\Delta U+P(V_2-V_1)$

$100=\Delta U+3\times {10}^3(400-200)\times {10}^{-6}$

$\therefore$ change in internal

energy $\Delta U=99.4J$

(b) by ideal gas equation

$P_1{V}_1=n_{1}R{T}_1$

$3\times {10}^3\times 200\times {10}^{-6}=n_1\times \frac{25}{3}\times 400$

$\therefore n_1=1.8\times {10}^{-4}moles$

(c) $Q=n{C}_p\left(dT\right)$ Heat supplied at constant pressure

Since pressure is constant

$\therefore \frac{V_1}{T_1}=\frac{V_2}{T_2}\therefore \frac{200}{400}=\frac{400}{72}$

$\therefore 72=800k$

$\therefore 100=1.8\times {10}^{-4}\times C_p\times (500-400)$

$\therefore C_p=1388.88\frac{J}{mol-k}$

Also,

d) $C_p-C_v=R$

$\therefore C_v=C_p-R$

$=1388.88-\frac{25}{3}$

$=1380.55\frac{J}{mol-k}.$