When an ideal diatomic gas is heated at constant pressure, we need to determine what fraction of the heat energy supplied goes into increasing the internal energy of the gas.
For a diatomic gas, the heat capacity at constant volume (Cv) and at constant pressure (Cp) are related by the following formulas:
1. [tex]\( Cv = \frac{5}{2} R \)[/tex]
2. [tex]\( Cp = Cv + R = \frac{5}{2} R + R = \frac{7}{2} R \)[/tex]
The fraction [tex]\( f \)[/tex] of the heat energy supplied that increases the internal energy of the gas is given by the ratio:
[tex]\[
f = \frac{Cv}{Cp}
\][/tex]
Substituting the values of [tex]\( Cv \)[/tex] and [tex]\( Cp \)[/tex] into the equation, we get:
[tex]\[
f = \frac{\frac{5}{2} R}{\frac{7}{2} R} = \frac{5}{7}
\][/tex]
Therefore, the fraction of the heat energy supplied that increases the internal energy of the gas is:
[tex]\[
\boxed{\frac{5}{7}}
\][/tex]
Considering the answer options provided:
A. [tex]\( \frac{2}{5} \)[/tex]
B. [tex]\( \frac{3}{5} \)[/tex]
C. [tex]\( \frac{3}{7} \)[/tex]
D. [tex]\( \frac{5}{7} \)[/tex]
The correct answer is [tex]\( \boxed{\frac{5}{7}} \)[/tex] which matches option D.
