Question

When a monatomic gas expands at constant pressure, the percentages of heat supplied that is used to do external work and to increase its internal energy are respectively

• A. \( 40, 60 \)
• B. \( 25, 75 \)
• C. \( 60, 40 \)
• D. \( 75, 25 \)

Hint:

For a monatomic ideal gas undergoing an isobaric process, use the first law of thermodynamics \( Q = \Delta U + W \). Recall the expressions for \( \Delta U \) and \( W \) in terms of \( n, R, \Delta T \), and the molar heat capacities \( C_v \) and \( C_p \). The ratio of work done to heat supplied and the ratio of change in internal energy to heat supplied will give the required percentages. For a monatomic gas, \( C_v = \frac{3}{2} R \) and \( C_p = \frac{5}{2} R \).

Answer 1

The Correct Option is A

For a monatomic ideal gas, the internal energy \( U \) is given by \( U = \frac{3}{2} nRT \), where \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature.
The change in internal energy \( \Delta U \) is \( \frac{3}{2} nR \Delta T \).
The heat supplied \( Q \) at constant pressure (isobaric process) is given by \( Q = n C_p \Delta T \), where \( C_p \) is the molar heat capacity at constant pressure.
For a monatomic ideal gas, \( C_p = \frac{5}{2} R \).
So, \( Q = n \left( \frac{5}{2} R \right) \Delta T = \frac{5}{2} nR \Delta T \).
The work done \( W \) by the gas during expansion at constant pressure is \( W = P \Delta V = nR \Delta T \) (from the ideal gas law \( PV = nRT \)).
Now, let's find the percentage of heat supplied used for external work: $$ \frac{W}{Q} \times 100\
= \frac{nR \Delta T}{\frac{5}{2} nR \Delta T} \times 100\
= \frac{1}{\frac{5}{2}} \times 100\
= \frac{2}{5} \times 100\
= 40\
$$ Next, let's find the percentage of heat supplied used to increase internal energy: $$ \frac{\Delta U}{Q} \times 100\
= \frac{\frac{3}{2} nR \Delta T}{\frac{5}{2} nR \Delta T} \times 100\
= \frac{\frac{3}{2}}{\frac{5}{2}} \times 100\
= \frac{3}{5} \times 100\
= 60\
$$ Therefore, the percentages of heat supplied used to do external work and to increase internal energy are 40
and 60
, respectively.