Internal energy and heat capacity

Question 1

dU=?

Answer 1

𝛿q + 𝛿w = 𝛿q-PdV

Question 2

Why do we write 𝛿 for q?

Answer 2

dq would imply a small change in the heat but heat is not a state function so we can’t say this

Question 3

In a constant volume process…

Give an example of this

Answer 3

dU = 𝛿q const vol eg. heat supplied to gas in sealed container, no work done as gas can’t expand so instead internal energy and heat increase

Question 4

For a substance with heat capacity c, q=?

Answer 4

cΔT= nC(m)ΔT

Question 5

What is C(m) and its unit?

Answer 5

Molar heat capacity, heat to raise one mole of substance by 1 degree, in J per K per mol

Question 6

At constant vol what is ΔU in terms of molar heat

capacity?

Answer 6

ΔU = nC(v,m)ΔT

Question 7

dU(m) (change in molar internal energy) =?

Answer 7

C(v,m) dT

Question 8

Therefore C(v,m)=?

Answer 8

(∂U(∂m)/∂T) const v

Question 9

What does the curly d indicate?

Answer 9

That only variation of U with T is being considered

Question 10

What is the definition of enthalpy?

Answer 10

H=U + PV, heat supplied to gas at constant external pressure

Question 11

What is dH at constant pressure?

Answer 11

𝛿q (const p)

Question 12

dH(m)=? (at constant pressure)

Answer 12

C(p,m) dT

Question 13

C(p,m)=?

Answer 13

Cp,m =
(∂Hm/∂T)
p

Question 14

What is an equation for variation of enthalpy with temperature?

Answer 14

H(m)(T2) - Hm(T1) = C(p,m) (T2-T1)

Question 15

How can heat capacity be measured?

Answer 15

see notes

Question 16

What is the parameterised form of heat capacities?

Answer 16

C(T) = a + bT + c/(T^2)

Question 17

Derive the expression for absolute entropy at constant pressure

Answer 17

see notes

Question 18

Definition of G?

Answer 18

G= H-TS

Question 19

Derive why if dG is negative, process is spontaneous

What are conditions for this derivation?

Answer 19

dG=dH -TdS+VdP
At constant temp dG/dT =-dH/dT + dS
ΔS(univ)= ΔS(surr) + ΔS(sys) = -q(sys)/T(surr) + ΔS(sys)
At constant P : dq=dH
dS(univ) =dH(sys)/dT + dS(sys)
So -dG(sys)=dS(univ)
Conditions: 1) Constant pressure 2) system and surroundings same temp, which is constant

Question 20

What are the 3 master equations?

Answer 20

1) dU = TdS - PdV
2) dH = TdS + VdP
3) dG = VdP - SdT
All of these are true under any conditions

Question 21

Derive expression for how G changes with pressure and volume (at constant temperature!)

Answer 21

use 3rd master equation : dG= VdP-SdT
at constant temp ∂G/∂P=V
dG=(nRT/P) dP
G(p2)-G(p1) =nRTln(p2/p1)
at p=1 bar, and using molar Gibbs energy
Gm(P) = G0m + RT ln (p/p0) T constant
V inversely proportional to P at constant T
so G(v2)-G(v1) = nRT ln(v1/v2)

Question 22

How does Gibbs energy vary with temp at constant pressure?

Answer 22

dG=-SdT constant pressure

(∂G/∂T)=-S constant P

Question 23

What is the Gibbs Helmholtz equation?

What are conditions for it to be true?

Answer 23

Want d/dT(G/T) at constant pressure
use product rule d/dt(G/T)= (1/T*dG/dT) - (G/T^2)
=-S/T - (H-TS)/ T^-2
= -H/T^2

Question 24

What does gibbs energy of each gaseous species in a gas mixture depend on?

Answer 24

Its partial pressure as by definition molecules in an ideal gas don’t interact

Question 25

p(i)=?

Answer 25

n(i)/n(tot)*p(tot)

Question 26

Therefore what is molar Gibbs energy of substance i at partial pressure pi?

Answer 26

see notes

Question 27

What is chemical potential?

Answer 27

Equivalent to molar Gibbs energy of an ideal gas but can be applied to ideal gases and solutions

Question 28

Write out chemical potential of a gas and a solution

Answer 28

see notes

Question 29

Why is the solution one not strictly accurate?

Answer 29

It technically only applies to ideal solutions with no significant interactions between solute molecules which is rarely true

Question 30

Solids and liquids chemical potential?

Answer 30

Always standard chemical potential at standard state