Question 520: Thermodynamics & KTG

Question

The temperature of 1 mole of an ideal monoatomic gas is increased by $50^{\circ} \mathrm{C}$ at constant pressure. The total heat added and change in internal energy are $E_1$ and $E_2$, respectively. If $\frac{E_1}{E_2}=\frac{x}{9}$ then the value of $x$ is _________.

Accepted Answer

$$ Q = n\,C_P\,\Delta T \quad 	ext{and} \quad \Delta U = n\,C_V\,\Delta T $$
For a monoatomic ideal gas, the molar specific heats are given by:
$$ C_V = \frac{3}{2}R \quad 	ext{and} \quad C_P = C_V + R = \frac{5}{2}R $$
Thus, the ratio of the heat added to the change in internal energy is:
$$ \frac{Q}{\Delta U} = \frac{C_P}{C_V} = \frac{\frac{5}{2}R}{\frac{3}{2}R} = \frac{5}{3} $$
Given that:
$$ \frac{E_1}{E_2} = \frac{x}{9} $$
We equate the two ratios:
$$ \frac{5}{3} = \frac{x}{9} $$
Multiplying both sides by 9:
$$ x = \frac{5}{3} 	imes 9 = 15 $$
Thus, the value of $$x$$ is
15
.

Thermodynamics & KTG

The temperature of 1 mole of an ideal monoatomic gas is increased by 50^{\circ} \mathrm{C} at constant pressure. The total heat added and change in internal energy are E_1 and E_2, respectively. If \frac{E_1}{E_2}=\frac{x}{9} then the value of x is _________.

Integer / Numeric Answer Type