(Molar specific heat capacity at constant pressure) – (Molar specific heat capacity at constant volume) is equal to
(A) R
(B) P
(C) V
(D) T
Answer
279.9k+ views
Hint: To understand and solve this question, we will have to first understand the molar specific heat capacity at constant pressure and at constant volume. We will also use the relationship between them to finally get our solution.
Complete answer:
First let us understand what molar specific heat capacity is at constant pressure and at constant volume.
Molar specific heat capacity at constant pressure is defined as the amount of heat energy required to raise the temperature of one mole of a substance through $ 1K $ or $ 1^\circ C $ at constant pressure. It is denoted by $ {C_P} $ .
By definition, $ {C_P} = \dfrac{{d{q_P}}}{{dT}} $
Since, $ {q_P} = H $
Therefore, $ {C_P} = \dfrac{{dH}}{{dT}} $
Molar specific heat capacity at constant volume is defined as the amount of heat energy required to raise the temperature of one mole of a substance through $ 1K $ or $ 1^\circ C $ at constant volume. It is denoted by $ {C_V} $ .
By definition, $ {C_V} = \dfrac{{d{q_V}}}{{dT}} $
Since, $ {q_V} = U $
Therefore, $ {C_V} = \dfrac{{dU}}{{dT}} $
Now let’s see the relationship between molar specific heat capacity at constant pressure and at constant volume.
From the equation of enthalpy, $ H = U + PV $
But, in case of one mole of ideal gas, $ PV = RT $
Therefore, $ H = U + RT $
On differentiating the both sides with respect to temperature, we get,
$ \dfrac{{dH}}{{dT}} = \dfrac{{dU}}{{dT}} + \dfrac{{d(RT)}}{{dT}} $
But we know that, $ {C_P} = \dfrac{{dH}}{{dT}} $ and $ {C_V} = \dfrac{{dU}}{{dT}} $
Therefore,
$ {C_P} = {C_V} + R.\dfrac{{dT}}{{dT}} $
$ {C_P} = {C_V} + R $
Or we can say, $ {C_P} - {C_V} = R $
Thus, the difference between the molar heat capacities at constant volume and pressure always equals $ R $ , the universal gas constant.
Hence, option A is correct.
Note:
In exams, you need not to memorize the whole thing. You can just remember the relationship between molar specific heat capacity at constant volume and at constant pressure for one mole, that is, $ {C_P} = {C_V} + R $ . You can also derive the relation between the two by simply using the enthalpy equation.
Complete answer:
First let us understand what molar specific heat capacity is at constant pressure and at constant volume.
Molar specific heat capacity at constant pressure is defined as the amount of heat energy required to raise the temperature of one mole of a substance through $ 1K $ or $ 1^\circ C $ at constant pressure. It is denoted by $ {C_P} $ .
By definition, $ {C_P} = \dfrac{{d{q_P}}}{{dT}} $
Since, $ {q_P} = H $
Therefore, $ {C_P} = \dfrac{{dH}}{{dT}} $
Molar specific heat capacity at constant volume is defined as the amount of heat energy required to raise the temperature of one mole of a substance through $ 1K $ or $ 1^\circ C $ at constant volume. It is denoted by $ {C_V} $ .
By definition, $ {C_V} = \dfrac{{d{q_V}}}{{dT}} $
Since, $ {q_V} = U $
Therefore, $ {C_V} = \dfrac{{dU}}{{dT}} $
Now let’s see the relationship between molar specific heat capacity at constant pressure and at constant volume.
From the equation of enthalpy, $ H = U + PV $
But, in case of one mole of ideal gas, $ PV = RT $
Therefore, $ H = U + RT $
On differentiating the both sides with respect to temperature, we get,
$ \dfrac{{dH}}{{dT}} = \dfrac{{dU}}{{dT}} + \dfrac{{d(RT)}}{{dT}} $
But we know that, $ {C_P} = \dfrac{{dH}}{{dT}} $ and $ {C_V} = \dfrac{{dU}}{{dT}} $
Therefore,
$ {C_P} = {C_V} + R.\dfrac{{dT}}{{dT}} $
$ {C_P} = {C_V} + R $
Or we can say, $ {C_P} - {C_V} = R $
Thus, the difference between the molar heat capacities at constant volume and pressure always equals $ R $ , the universal gas constant.
Hence, option A is correct.
Note:
In exams, you need not to memorize the whole thing. You can just remember the relationship between molar specific heat capacity at constant volume and at constant pressure for one mole, that is, $ {C_P} = {C_V} + R $ . You can also derive the relation between the two by simply using the enthalpy equation.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are

Which of the following would not be a valid reason class 11 biology CBSE

Why should electric field lines never cross each other class 12 physics CBSE

An electrostatic field line is a continuous curve That class 12 physics CBSE

What is meant by monosporic development of female class 11 biology CBSE

Draw labelled diagram of the following i Gram seed class 11 biology CBSE

Trending doubts
What is 1 divided by 0 class 8 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is the past tense of read class 10 english CBSE

What is pollution? How many types of pollution? Define it

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

How many crores make 10 million class 7 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE
