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The entropy of a system of N spins, which may align either in the upward or in the downward direction. The probability of upward directions is 1/3 and that of downward direction is 2/3. The entropy per spin in unit of kB is _______
Correct answer is '0.64'. Can you explain this answer?
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Entropy of a System of N Spins
The entropy of a system measures the degree of disorder or randomness in the system. In the context of spins, it quantifies the number of possible spin configurations that the system can have.
Probability of Upward and Downward Directions
Given that the probability of upward directions is 1/3 and that of downward direction is 2/3, we can calculate the entropy per spin in units of kB.
Calculating the Entropy
The entropy per spin can be calculated using the formula:
Entropy per spin = - (p1 * log2(p1) + p2 * log2(p2) + ... + pN * log2(pN))
where p1, p2, ..., pN are the probabilities of each possible spin configuration. In this case, we have two possible spin configurations: upward and downward.
Let's calculate the entropy per spin using the given probabilities:
Entropy per spin = - (1/3 * log2(1/3) + 2/3 * log2(2/3))
Calculating each term separately:
Term 1: (1/3 * log2(1/3))
Using log2(1/3) = -log2(3), we have:
Term 1 = (1/3 * -log2(3))
Term 2: (2/3 * log2(2/3))
Using log2(2/3) = -log2(3/2), we have:
Term 2 = (2/3 * -log2(3/2))
Now, substituting the calculated values back into the entropy per spin formula:
Entropy per spin = - (1/3 * -log2(3) + 2/3 * -log2(3/2))
Simplifying further:
Entropy per spin = log2(3) - (2/3 * log2(3/2))
Using log2(3/2) = log2(3) - log2(2), we have:
Entropy per spin = log2(3) - (2/3 * (log2(3) - log2(2)))
Simplifying again:
Entropy per spin = log2(3) - (2/3 * log2(3)) + (2/3 * log2(2))
Combining like terms:
Entropy per spin = (1 - 2/3) * log2(3) + (2/3 * log2(2))
Simplifying further:
Entropy per spin = (1/3) * log2(3) + (2/3 * log2(2))
Calculating the values:
Entropy per spin ≈ 0.64
Therefore, the entropy per spin in units of kB is approximately 0.64.
The entropy of a system measures the degree of disorder or randomness in the system. In the context of spins, it quantifies the number of possible spin configurations that the system can have.
Probability of Upward and Downward Directions
Given that the probability of upward directions is 1/3 and that of downward direction is 2/3, we can calculate the entropy per spin in units of kB.
Calculating the Entropy
The entropy per spin can be calculated using the formula:
Entropy per spin = - (p1 * log2(p1) + p2 * log2(p2) + ... + pN * log2(pN))
where p1, p2, ..., pN are the probabilities of each possible spin configuration. In this case, we have two possible spin configurations: upward and downward.
Let's calculate the entropy per spin using the given probabilities:
Entropy per spin = - (1/3 * log2(1/3) + 2/3 * log2(2/3))
Calculating each term separately:
Term 1: (1/3 * log2(1/3))
Using log2(1/3) = -log2(3), we have:
Term 1 = (1/3 * -log2(3))
Term 2: (2/3 * log2(2/3))
Using log2(2/3) = -log2(3/2), we have:
Term 2 = (2/3 * -log2(3/2))
Now, substituting the calculated values back into the entropy per spin formula:
Entropy per spin = - (1/3 * -log2(3) + 2/3 * -log2(3/2))
Simplifying further:
Entropy per spin = log2(3) - (2/3 * log2(3/2))
Using log2(3/2) = log2(3) - log2(2), we have:
Entropy per spin = log2(3) - (2/3 * (log2(3) - log2(2)))
Simplifying again:
Entropy per spin = log2(3) - (2/3 * log2(3)) + (2/3 * log2(2))
Combining like terms:
Entropy per spin = (1 - 2/3) * log2(3) + (2/3 * log2(2))
Simplifying further:
Entropy per spin = (1/3) * log2(3) + (2/3 * log2(2))
Calculating the values:
Entropy per spin ≈ 0.64
Therefore, the entropy per spin in units of kB is approximately 0.64.
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The entropy of a system of N spins, which may align either in the upward or in the downward direction. Theprobability of upward directions is 1/3and that of downward direction is 2/3.The entropy per spin in unit of kBis _______Correct answer is '0.64'. Can you explain this answer?
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