The temperature of 1 mole of an ideal monoatomic gas is increased by \( 50^\circ {C} \) 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:
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Approach Solution - 1
Given:
- The process is isobaric, and \( \Delta T = 50^\circ \text{C} \). - The heat added in an isobaric process is \( Q = n C_p \Delta T = E_1 \). - The change in internal energy in an isobaric process is \( \Delta U = n C_v \Delta T = E_2 \).
Step 1: Relating the energies and the heat capacities
Since \( \frac{E_1}{E_2} = \frac{C_p}{C_v} = \gamma \), we can relate the ratio of the heat capacities to the ratio of the energies. \[ \frac{E_1}{E_2} = \frac{C_p}{C_v} = \gamma. \]
Step 2: Use the specific heat capacities for a monoatomic gas
For a monoatomic gas, the value of \( \gamma \) is given by: \[ \gamma = 1 + \frac{2}{f}, \] where \( f \) is the number of degrees of freedom of the gas. For a monoatomic gas, \( f = 3 \). Substituting this value: \[ \gamma = 1 + \frac{2}{3} = \frac{5}{3}. \]
Step 3: Solving for \( x \)
The equation given in the problem is: \[ \frac{5}{3} = \frac{x}{9}. \] Solving for \( x \): \[ x = 15. \]
Final Answer:
The value of \( x \) is \( \boxed{15} \).
Approach Solution -2
Given: The process is isobaric, meaning that the temperature change \( \Delta T = 50^\circ C \). The heat in an isobaric process is given by: \[ Q = n C_p \Delta T = E_1, \] and the change in internal energy in an isobaric process is: \[ \Delta U = n C_v \Delta T = E_2. \] Now, the ratio of these quantities is: \[ \frac{E_1}{E_2} = \frac{C_p}{C_v} = \gamma. \]
Step 1: Given that the gas is monoatomic
For a monoatomic gas, the specific heat ratio \( \gamma \) is given by: \[ \gamma = 1 + \frac{2}{f}. \] Where \( f \) is the degrees of freedom of the gas. For a monoatomic gas, \( f = 3 \), so: \[ \gamma = 1 + \frac{2}{3} = \frac{5}{3}. \]
Step 2: Solve for \( x \)
We are given the equation: \[ \frac{5}{3} = \frac{x}{9}. \] Solving for \( x \): \[ x = \frac{5}{3} \times 9 = 15. \]
Final Answer:
The value of \( x \) is \( \boxed{15} \).
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