Question

Heat energy of \[{{743 J}}\] is needed to raise the temperature of \[{{5}}\] moles of an ideal gas by \[{{2K}}\] at constant pressure. How much heat energy is needed to raise the temperature of the same mass of the gas by \[{{2K}}\] at constant volume?

Answer 1

Hint: Heat energy is the form of energy transfer. The transferring this energy occurs in between two different systems. By using the relation between heat energy and volume at constant pressure and pressure and volume we can solve this problem.

Complete step by step answer:
The relation between heat energy and volume at constant pressure, is given by:
${{Q = n}}{{{C}}_{{p}}}{{\Delta T}}$
Where,
${{\text{Q = heat energy,}}}$
${{\text{n = number of moles,}}}$
${{\text{Cp = specific heat at constant pressure}}}$
${{T = temperature}}$
From the given,
Heat energy = \[{{743J}}\]
${{{Q}}_{{p}}}{{ = n}}{{{C}}_{{p}}}{{\Delta T = 743J}}$……………………………. (1)
The relation between heat energy and volume at constant volume, is given by:
${{{Q}}_{{c}}}{{ = n}}{{{C}}_{{v}}}{{\Delta T = n(}}{{{C}}_{{p}}}{{ - R)}}\partial {{T}}$
${{{Q}}_{{c}}}{{ = n}}{{{C}}_{{p}}}\Delta {{T - nR}}\partial {{T}}$……………………………. (2)
Where,
${{\text{Cv = specific heat at constant volume}}}$
${{\text{R = gas constant = 8}}{{.314 J mo}}{{{l}}^{{{ - 1}}}}{{{K}}^{{{ - 1}}}}}$
\[{\text{n = Number of moles = 5}}\]
\[{{T = Temperature = 2K}}\]
From the given,
Therefore, on substituting the values in equation (2).
${{{Q}}_{{c}}}{{ = 743 - (5 \times 8}}{{.314 \times 2)}}$
${{ = 660 J}}$
Hence, the heat energy at constant volume is obtained as 660 joules.

Additional Information:
-The temperature of the system increases when heat is consumed by the system. And, the temperature is reduced when heat is lost.
-An entity 's temperature is the measurement of the overall kinetic energy of the molecules that constitute that entity. So, this heat is transformed into the kinetic energy of the particles when heat is absorbed by an entity and this results in temperature increase.

Note:
The heat required to increase the temperature by one degree Celsius per unit mass is defined as specific heat. When a phase transition is observed, the relationship does not apply, because the heat added or removed during a phase transition does not alter the temperature.