Question

explain me the relationship between cp and cv for an ideal gas??

Answer 1

Relationship between C_v and C_p for an ideal gas is as follows:

Heat capacity, C=q/T [where, q: Heat; T: Temperature]

At constant volume, Heat capacity is denoted by C_v.

So, C_v= q_v/∆T

Or, q_v=C_v∆T -----------------(1)

Since, ∆U=q - P∆V [Where, ∆U: Internal energy change]

At constant volume , P∆V=0

Therefore, at constant volume, ∆U=q_v -------(2)

From (1) and (2),

∆U =C_v∆T

At constant pressure, Heat capacity is denoted by C_p.

So, C_p= q_p/∆T

q_p= C_p∆T ---------------------(3)

we know, ∆H=∆U+∆(PV) [where, ∆H:enthalpy change]

At constant pressure, ∆H=∆U+ P∆V

Since at constant pressure, ∆U=q_p- P∆V

q_p=∆U+P∆V

q_p=∆H -------------(4)

From (3) and (4),

∆H= C_p∆T

Since, ∆H=∆U+∆(PV)

∆H=∆U+∆(nRT) [we are deriving Relationship between C_v and C_p for an ideal gas]

∆H=∆U+∆(RT) [For 1 mole of an ideal gas]

∆H=∆U+R∆T

After putting the values of ∆H and ∆U in above equation ,we get,

C_p∆T = C_v∆T + R∆T

C_p = C_v + R or C_p- C_v = R