Gibbs Free Energy

This is another thermodynamic quantity that helps in predicting the spontaneity of a process, is called Gibbs energy (G).  

Gibbs free energy is the energy needed to create a system out of nothing in an environment at both constant pressure and constant temperature.

It is defined mathematically by the equation.  

G = H - TS

Where H = heat content, S = entropy of the system, T = absolute temperature  

Solved example: Which of the following will fit into the blank?  

When two phases of the same single substance remain in equilibrium with one another at a constant P and T, their molar _________ must be equal.  

             (A) Internal energy                              (B) Enthalpy  

             (C) Entropy                                         (D) Free energy  

Solution: (D)    

Free energy change  

For isothermal process.  

      ΔG = G₂ – G₁ = ΔH – TΔS  

ΔG = change in Gibbs free energy of the system.  

It is that thermodynamic quantity of a system the decrease in whose value during a process is equal to the maximum possible useful work that can be obtained from the system.  

Solved example: Calculate free energy change when one mole of NaCl is dissolved in water at 25°C. Lattice energy = 700 kJ/mol.ΔS at 25°C = 26.5 Jmol^–1, Hydration energy of NaCl = -696 kJ/mol.  

                        (A) -3.9 kJ                                         (B) -8kJ  

                        (C) -12kJ                                           (D) -16kJ    

Solution: (A)

Problem 1:-

Calculate the total rotational kinetic energy of all the molecules in one mole of air at 25.0ºC.

Concept:

Let us assume all the molecules in the air are diatomic. For half rotation the molecule will contribute ½ kT to its rotational kinetic energy, where k is the Boltzmann’s constant. So for one complete rotation the molecule will contribute kT (½ kT×2= kT) to its rotational kinetic energy.

So the total rotational kinetic energy will be,

E = kT

But k= R/N_A

Where R is the gas constant and N_A is the Avogadro’s number.

To find out total rotational kinetic energy E, substitute R/N_A for k in the equation E = kT,

E = kT

   = (R/N_A) T  

For one mole of gas (N_A =1), the equation E =(R/N_A)T  will be,

E = RT      

To find out the total rotational kinetic energy of all the molecules, substitute 25.0^° C for temperature T and 8.31 J/ K for R for one mole of gas in the equation E = RT,

E = RT

   = (8.31 J/ K) (25.0^° C)

  = (8.31 J /K)(25+273) K

  = (8.31 J/ K)(298 K)

 = 2476.38 J

Rounding off to three significant figure, the total rotational kinetic of all the molecule will be 2480 J.

Problem 2:-

Calculate the internal energy of 1 mole of an ideal gas at 250ºC.

Concept:

Internal energy (E_int) of the ideal gas is defined as the,

E_int = 3/2 nRT

Where n is the number of moles, R is the gas constant and T is the temperature.

To find out the internal energy of the ideal gas, substitute 1 mole for n, 250^° C for temperature T and 8.31 J/ mol. K for the gas constant R in the equation E = 3/2 nRT,

E = 3/2 nRT

   = 3/2 (1 mol) (8.31 J/ mol. K) (250^° C)

  = (1.5)( 1 mol) (8.31 J/ mol. K) (250+273) K

  = (1.5)( 1 mol) (8.31 J/ mol. K) (523) K

 = 6519.195 J

Rounding off to three significant figures, the internal energy of the ideal gas will be 6520 J.

Problem 3:-

A cosmic-ray particle with energy 1.34 TeV is stopped in a detecting tube that contains 0.120 mpl of neon gas. Once this energy is distributed among all the atoms, by how much is the temperature of the neon increased? 

Concept:

We can assume neon is an ideal gas. The internal energy (ΔE_int) of the ideal gas is defined as the,

ΔE_int = 3/2 nR ΔT

Where n is the number of moles, R is the gas constant and ΔT is the increase in temperature.

From the equation ΔE_int = 3/2 nR ΔT, the increase in temperature ΔT will be,

ΔT = 2/3 ΔE_int/nR

To obtain the increase in temperature ΔT, substitute 1.34 TeV for ΔE_int, 0.120 mol for number of mole n and 8.31 J/ mol. K for the gas constant R in the equation ΔT = 2/3 ΔE_int/nR,

ΔT = 2/3 ΔE_int/nR

     = 2/3 (1.34 TeV) (10¹² eV/1 TeV) (1.6×10^-19 J/eV) / (0.120 mol) (8.31 J/ mol. K)

     = 1.43×10^-7 K

From the above observation we conclude that, the increased temperature of the neon will be 1.43×10^-7 K.

Problem 4:-

A mixture of 1.78 kg of water and 262 g of ice at 0ºC is, in a reversible process, brought to a final equilibrium state where the water/ice ratio, by mass, is 1:1 at 0ºC. (a) Calculate the entropy change of the system during this process. (b) The system is then returned to the first equilinrium state, but in an reversible way. Calculate the entropy change of the system during this process. (c) Show that your answer is consistent with the second law of thermodynamics.

Concept:

The entropy change ΔS for a reversible isothermal process is defined as,

ΔS = Q/T

     = -mL/T

Here m is the mass, L is the latent heat and T is the temperature.

1. Mass of water = 1.78 kg

Mass of ice = 262 g

So the total mass of ice and water mixture will be,

Mass of ice-water mixture = (Mass of water) + (Mass of ice)

                                            = (1.78 kg) + (262 g)

                                            = (1.78 kg) + (262 g×10^-3 kg/1 g)

                                            = 1.78 kg + 0.262 kg

                                            = 2.04 kg

If eventually the ice and water have the same mass, then the final state will have 1.02 kg (2.04 kg/2) of each.

Thus the mass of the water that changed into ice m will be the difference of mass of water m_w  and mass of final state m_s.

So, m = m_w - m_s

m = m_w - m_s

   = 1.78 kg – 1.02 kg

  = 0.76 kg

The change of water at 0^° C to ice at 0^° C  is isothermal.

ΔS = -mL/T

     = -(0.76 kg) (333×10³ J/kg )/(273 K)

     = -927 J/K

From the above observation we conclude that, the change in entropy ΔS of the system during this process will be -927 J/K.

2. Now the system is returned to the first equilibrium state, but in an irreversible way. Thus the change in entropy ΔS of the system during this process is equal to the negative of previous case.

So, ΔS = -(- 927 J/K)

           = 927 J/K

From the above observation we conclude that, the change in entropy ΔS of the system would be 927 J/K.

3. In accordance to second law of thermodynamics, entropy change ΔS is always      zero.

The total change in entropy will be,

ΔS = (-927 J/K) + (927 J/K)

      = 0

From the above observation we conclude that, our answer is consistent with the second law of thermodynamics.

Problem 5:-

Water standing in the open at 32ºC evaporates because of the escape of some of the surface molecules. The heat vaporization is approximately equal to εn, where ε is the average energy of the escaping molecules and n is the number of molecules per kilogram. (a) Find ε. (b) What is ratio of ε to the average kinetic energy of H₂O molecules, assuming that the kinetic energy is related to temperature in the same way as it is for gases?

Concept:

Number of molecules n is equal to the Avogadro’s number N_A divided by molecular weight M.

So, n = N_A/M

The average kinetic energy E_av of a diatomic molecule like H₂O is defined as,

E_av = 3/2 kT

Here k is the Boltzmann constant and T is the temperature.

Solution:

(a)

As heat of vaporization L_v is approximately equal to εn, where ε is the average energy of the escaping molecules and n is the number of molecules per kilogram, so,

L_v = εn

Or,

ε = L_v/n

   = L_v/( N_A/M)

   = L_vM / N_A

To obtain the average energy ε, substitute 2256×10³ J/kg for latent heat of vaporization of water molecule (H₂O) L_v, 0.018 kg/mol for M and 6.03×10²³/mol for N_A in the equation ε = L_vM / N_A, we get,

ε = L_vM / N_A

 = (2256×10³ J/kg)( 0.018 kg/mol)/( 6.03×10²³)

= 6.75×10^-20 J

From the above observation we conclude that, the value of average energy ε would be 6.75×10^-20 J.

(b)

As, E_av = 3/2 kT, therefore the ratio of ε to the average kinetic energy E_av of H₂O molecule will be,

ε / E_av = ε / (3/2 kT)

           = 2 ε / 3 kT

To obtain the ratio of ε to the average kinetic energy E_av of H₂O molecule, substitute 6.75×10^-20 J for ε , 1.38×10^-23 J/K for k and 32° C for T in the equation ε / E_av= 2 ε / 3 kT, we get,

ε / E_av= 2 ε / 3 kT

           = 2(6.75×10^-20 J)/3(1.38×10^-23 J/K)( 32° C)

           = 2(6.75×10^-20 J)/3(1.38×10^-23 J/K)( 32+273) K)

          =2(6.75×10^-20 J)/3(1.38×10^-23 J/K)( 305K)

          = 10.7

From the above observation we conclude that, the ratio of ε to the average kinetic energy E_av of H₂O molecule would be 10.7.

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