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Change in accessible states relating to change in energy

  1. Oct 26, 2014 #1
    1. The problem statement, all variables and given/known data
    A certain system has 6 × 10^24 degrees of freedom. Its internal energy
    increases by 1%. By what factor does the number of accessible states increase?

    2. Relevant equations
    [tex]\Omega = E^{N\nu/2}[/tex]
    [itex]\nu[/itex] is the degrees of freedom, and N is just 1, so we can ignore that. So the exponent is just 3x10^24
    [itex]\Omega[/itex] is the number of accessible states.
    3. The attempt at a solution

    First I replaced E with 10. And then E increased by 1% would be 10.1.
    What I was going to try to do was divide 10.1^(6x10^24) by 10^(6x10^24) to get my answer. But no calculator in the world can do that.
    So I used a calculator that can do big numbers (but not quite that big). I found that as I increased the 0's in the exponent, the exponent of my answer increased by some seemingly random number. I tried 10.1^(300)/10^(300) and then 10.1^(3000)/10^(3000), and then kept adding zeroes like that to see what kind of pattern I got.
    What I got was, as I got up to 10 zeroes, was an answer of 3x10^(129641213). If I take away a zero from the exponents in the fraction, then I just lose an exponent in the answer. So if I had 24 zeroes in my exponents in my fraction, then the answer would be roughly 3x10^(1.29x10^22).

    But this can't be the way to do this problem. Anyone have an alternative idea on how to solve this? Thanks.
  2. jcsd
  3. Oct 26, 2014 #2


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    Try taking logarithms on both sides: the desired ratio is ##\Omega '/\Omega = \exp ...##
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