### Boltzmann Factor Calculator :

Enter the E₁ in electron volts(eV), E₂ in electron volts(eV), and Temperature in celsius(°C), and you get below the calculator.

### Boltzmann Formula :

Boltzmann formula is the probability at which that this sate occurs P_{1} / P_{2 } is equal to the exp and multiply the E1 is the energy of the state 1 E_{1(j)} in joules and minutes the E2 is the Energy of the state 2 E_{2(j)} in joules and divided by the Boltzmann constant KB_{(j/k)} in joule per kilo and multiply the temperature T_{(k)} in kelvins. Hence the Boltzmann formula can be written as

P_{1} / P_{2} = exp (- (E_{1(j)} – E_{2(j)} ) / (KB_{(j/k)} *T_{(k)} )

### Derivation :

Boltzmann Distribution or Gibbs distribution

P=1/Z * exp (-E /( KB *T )

z = normalization constant

E= Energy of state in joules

KB =1.38065 *10 ^(-23 ) J/K is the Boltzmann constant

T= temperature in kelvins

P=probability at which that this sate occurs

Boltzmann factor tells us the relative probability with which two states of Energies E_{1} and E_{2} occurs. Dividing the Boltzmann Distribution for these two states we get

P_{1} / P_{2} =exp (- (E_{1} – E_{2} ) / (KB *T )

It is evident that relative probability depends on the difference of energies. The other factor that plays a vital role in Boltzmann Factor is temperature. The Lower the Temperature is the more probable is the state of lower Energy

### Boltzmann factor Example: 1

Calculate the probability value? Energy of state 9 joules and 2^{nd} energy state 9 joules and temperature 5K .

### Solution:

E_{1}=9 joules

E_{2}=9 joules

T=5k

By using formula

P_{1} / P_{2} = exp (- (E_{1} – E_{2} ) / (KB *T )

P_{1} / P_{2} = exp (9-9 ) / (1.38065 *10^{-23} * 5)

Boltzmann factor =1

### Boltzmann Factor Example: 2

Calculate the probability value? Energy of state 35 joules and 2^{nd} energy state 35 joules and temperature 12K .

### Answer:

E_{1}=35 joules

E_{2}=35 joules

T=12 k

By using formula

P_{1} / P_{2} = exp (- (E_{1} – E_{2} ) / (KB *T )

P_{1} / P_{2} = exp (35 – 35 ) / (1.38065 *10^{-23} * 12)

Boltzmann factor =1