The probability of occupying the ground level increases as the temperature increases, while the probability of occupying the excited states decreases.
Given data: Nondegenerate energy levels with ε/k = 0, 100, and 200 K.'
The probability of occupying the ground level (i=0) when T=90 K is:
P0,90K = e^(-ε0/kT) / { e^(-ε0/kT) + e^(-ε1/kT) + e^(-ε2/kT) }
MP0,90K = e^(-0/k × 90 K) / { e^(-0/k × 90 K) + e^(-100/k × 90 K) + e^(-200/k × 90 K) }P0,90K
= 1 / { 1 + e^(-100 × 9) + e^(-200 × 9) }= 0.9475 (approximately)
The probability of occupying the excited state (i=1)
when T=90 K is:P1,90K = e^(-ε1/kT) / { e^(-ε0/kT) + e^(-ε1/kT) + e^(-ε2/kT) }P1,90K
= e^(-100/k × 90 K) / { e^(-0/k × 90 K) + e^(-100/k × 90 K) + e^(-200/k × 90 K) }P1,90K
= e^(-9000) / { 1 + e^(-9000) + e^(-18000) }= 0.052 (approximately)
The probability of occupying the excited state (i=2) when
T=90 K is:P2,90K = e^(-ε2/kT) / { e^(-ε0/kT) + e^(-ε1/kT) + e^(-ε2/kT) }P2,90K
= e^(-200/k × 90 K) / { e^(-0/k × 90 K) + e^(-100/k × 90 K) + e^(-200/k × 90 K) }P2,90K
= e^(-18000) / { 1 + e^(-9000) + e^(-18000) }
= 0.0005 (approximately)
The probability of occupying the ground level (i=0) when
T=900 K is:
P0,900K = e^(-ε0/kT) / { e^(-ε0/kT) + e^(-ε1/kT) + e^(-ε2/kT) }
P0,900K = e^(-0/k × 900 K) / { e^(-0/k × 900 K) + e^(-100/k × 900 K) + e^(-200/k × 900 K) }
P0,900K = 1 / { 1 + e^(-100 × 90) + e^(-200 × 90) }
= 0.9999999999970 (approximately)
The probability of occupying the excited state (i=1)
when T=900 K is:P1,900K = e^(-ε1/kT) / { e^(-ε0/kT) + e^(-ε1/kT) + e^(-ε2/kT) }P1,900K
= e^(-100/k × 900 K) / { e^(-0/k × 900 K) + e^(-100/k × 900 K) + e^(-200/k × 900 K) }
P1,900K = e^(-90000) / { 1 + e^(-90000) + e^(-180000) }= 1.5 × 10^-8 (approximately)
Therefore, the probability of occupying the ground level increases as the temperature increases, while the probability of occupying the excited states decreases.
To know more about Probability, The probability of occupying the ground level increases as the temperature increases, while the probability of occupying the excited states decreases.
Given data: Nondegenerate energy levels with ε/k = 0, 100, and 200 K.'
The probability of occupying the ground level (i=0) when T=90 K is:
P0,90K = e^(-ε0/kT) / { e^(-ε0/kT) + e^(-ε1/kT) + e^(-ε2/kT) }
MP0,90K = e^(-0/k × 90 K) / { e^(-0/k × 90 K) + e^(-100/k × 90 K) + e^(-200/k × 90 K) }P0,90K
= 1 / { 1 + e^(-100 × 9) + e^(-200 × 9) }= 0.9475 (approximately)
The probability of occupying the excited state (i=1)
when T=90 K is:P1,90K = e^(-ε1/kT) / { e^(-ε0/kT) + e^(-ε1/kT) + e^(-ε2/kT) }P1,90K
= e^(-100/k × 90 K) / { e^(-0/k × 90 K) + e^(-100/k × 90 K) + e^(-200/k × 90 K) }P1,90K
= e^(-9000) / { 1 + e^(-9000) + e^(-18000) }= 0.052 (approximately)
The probability of occupying the excited state (i=2) when
T=90 K is:P2,90K = e^(-ε2/kT) / { e^(-ε0/kT) + e^(-ε1/kT) + e^(-ε2/kT) }P2,90K
= e^(-200/k × 90 K) / { e^(-0/k × 90 K) + e^(-100/k × 90 K) + e^(-200/k × 90 K) }P2,90K
= e^(-18000) / { 1 + e^(-9000) + e^(-18000) }
= 0.0005 (approximately)
The probability of occupying the ground level (i=0) when
T=900 K is:
P0,900K = e^(-ε0/kT) / { e^(-ε0/kT) + e^(-ε1/kT) + e^(-ε2/kT) }
P0,900K = e^(-0/k × 900 K) / { e^(-0/k × 900 K) + e^(-100/k × 900 K) + e^(-200/k × 900 K) }
P0,900K = 1 / { 1 + e^(-100 × 90) + e^(-200 × 90) }
= 0.9999999999970 (approximately)
The probability of occupying the excited state (i=1)
when T=900 K is:P1,900K = e^(-ε1/kT) / { e^(-ε0/kT) + e^(-ε1/kT) + e^(-ε2/kT) }P1,900K
= e^(-100/k × 900 K) / { e^(-0/k × 900 K) + e^(-100/k × 900 K) + e^(-200/k × 900 K) }
P1,900K = e^(-90000) / { 1 + e^(-90000) + e^(-180000) }= 1.5 × 10^-8 (approximately)
Therefore, the probability of occupying the ground level increases as the temperature increases, while the probability of occupying the excited states decreases.
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