Question

If \( m_a \) and \( m_i \) are the slopes of the adiabatic and isothermal curves for an ideal gas, then \[ \left( \frac{c_p}{c_v} = \gamma \right) \]

• A. \( m_a = \gamma m_i \)
• B. \( m_i = \gamma m_a \)
• C. \( m_a m_i = \gamma \)
• D. \( m_a m_i = \gamma^2 \)
• E. \( \sqrt{\frac{m_a}{m_i}} = \gamma \)

Hint:

Remember that the ratio of specific heats (\(\gamma\)) plays a key role in relating the slopes of the adiabatic and isothermal curves for an ideal gas.

Answer 1

The Correct Option is A

For an ideal gas, the slopes of the adiabatic and isothermal curves are related to the specific heat capacities and the ratio of specific heats \( \gamma \). 
The general relationship between the slopes of the adiabatic and isothermal curves is given by: \[ \frac{c_p}{c_v} = \gamma \] The slope of the adiabatic curve \( m_a \) and the slope of the isothermal curve \( m_i \) are related through the ratio of specific heats \( \gamma \). 
From the ideal gas laws and thermodynamic relationships, we know that: \[ m_a = \gamma m_i \] 
Thus, the correct relationship between the slopes is \( m_a = \gamma m_i \).