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 _________.

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