Calcyulating number of protons and neutrons in early universe

[TFM]

Homework Statement

At very high temperatures (as in the early universe), the proton and the neutron can be thought of as two different states of the same particle, called the “nucleon”. (The reactions that convert a proton to a neutron or vice versa require the absorption of an electron or a positron or a neutrino, but all of these particles tend to be very abundant at sufficiently high temperatures.) Since the neutron’s mass is higher than the proton’s by 2.3 × 10−30 kg, its energy is higher by this amount times c2. Suppose, then, that at some very early time, the nucleons were in thermal equilibrium with the rest of the universe at 1011 K. What fraction of the nucleons at that time were protons, and what fraction were neutrons?

Homework Equations

$$\frac{P(s_2)}{P(s_1)} = \frac{e^-E(s_1)/k_BT}{e^-E(s_2)/k_BT}$$

The Attempt at a Solution

I have done most of this question.

I have said (s_1) is the neutron, and (s_2) = the proton

S_1 = 2.3 * 10^{-30}C^2 = 2.07*10^{-13}

S_2 = 0

I have inserted these values, as well as T = 10^11, into the above equation and have got:

$$\frac{P(s_2)}{P(s_1)} = \frac{1}{0.98} = 1.015$$

I have now arranged it to say:

P(protons) = 1.015 P(Neutrons)

from this, how can I calculate the fraction of the nucleons that were protons, and what fraction were neutrons?

Many Thanks,

TFM

 

[Delphi51]

Well, the part you did is beyond my knowledge, but I think I can help you with the last bit.
You found that the number of protons is 1.015 times the number of neutrons, right?
If x is the number of neutrons, then you have 1.015x protons. The fraction of the total that is neutrons is x/(x + 1.015x).

 

[TFM]

Would they want the answer in terms of x, or do I need to find out what x is?

 

[Vuldoraq]

I'm not sure you have gone about this the right way. You have calculated the ration of probabilties correctly, but they want the fraction of neutrons and protons. So you need something like,

P(s1)/[P(s1)+P(s2)] and P(s2)/[P(s1)+P(s2)]

 

[TFM]

so that will be:

$$\frac{e^{E(s_2}/k_BT}{E(s_2}/k_BT + E(s_1}/k_BT}$$

and

$$\frac{e^{E(s_1}/k_BT}{E(s_2}/k_BT + E(s_1}/k_BT}$$

Right?

 

[Vuldoraq]

Yeah, but those energies over kt should have exponentials. Also I have noticed there is an arithmetic error in your original calculation. For one of the energies you get 0.98, this is incorrect.

 

[TFM]

Okay, so:

$$P(N) = \frac{e^{E(s_1)/k_BT}}{e^{E(s_1)/k_BT} + e^{E(s_2)/k_BT}}$$

and:

$$P(P) = \frac{e^{E(s_2)/k_BT}}{e^{E(s_1)/k_BT} + e^{E(s_2)/k_BT}}$$Now:

for Proton:

$$e^{E(s_2)/k_BT}$$

E(s2) = 0

so

$$e^{0)/k_BT} = 1$$

Now for the neutron:

$$e^{E(s_1)/k_BT}$$

E = 2.3 * 10^30 * C^2 = 2.3 * 10^30 * (3 * 10^8)^2 = 2.07*10^-13

kB = 1.38*10^-23
T = 10^11

thus:

$$k_BT = 1.38 * 10^{-11}$$

so:

$$e^{E(s_1)/k_BT} = 0.015$$

So:

$$P(N) = \frac{e^{E(s_1)/k_BT}}{e^{E(s_1)/k_BT} + e^{E(s_2)/k_BT}} = \frac{0.015}{0.015 + 1} = \frac{0.015}{1.015} = \frac{15}{1015}$$

and

$$P(P) = \frac{e^{E(s_2)/k_BT}}{e^{E(s_1)/k_BT} + e^{E(s_2)/k_BT}} = \frac{1}{0.015 + 1} = \frac{1}{1.015} = \frac{1000}{1015}$$

Does this look right? There seems to be a lot more protons to neutrons?

Then again, The Early Universe would have been mostly hydrogen, which is just protons.

 

[Vuldoraq]

Okay, so:

$$P(N) = \frac{e^{E(s_1)/k_BT}}{e^{E(s_1)/k_BT} + e^{E(s_2)/k_BT}}$$

and:

$$P(P) = \frac{e^{E(s_2)/k_BT}}{e^{E(s_1)/k_BT} + e^{E(s_2)/k_BT}}$$


Now:

for Proton:

$$e^{E(s_2)/k_BT}$$

E(s2) = 0

so

$$e^{0)/k_BT} = 1$$

Fine

Now for the neutron:

$$e^{E(s_1)/k_BT}$$

E = 2.3 * 10^30 * C^2 = 2.3 * 10^30 * (3 * 10^8)^2 = 2.07*10^-13

kB = 1.38*10^-23
T = 10^11

thus:

$$k_BT = 1.38 * 10^{-11}$$

This should be 1.38*10^-12. If you make the correction here the rest should come out fine.

so:

$$e^{E(s_1)/k_BT} = 0.015$$

So:

$$P(N) = \frac{e^{E(s_1)/k_BT}}{e^{E(s_1)/k_BT} + e^{E(s_2)/k_BT}} = \frac{0.015}{0.015 + 1} = \frac{0.015}{1.015} = \frac{15}{1015}$$

and

$$P(P) = \frac{e^{E(s_2)/k_BT}}{e^{E(s_1)/k_BT} + e^{E(s_2)/k_BT}} = \frac{e^{1}{0.015 + 1} = \frac{1}{1.015} = \frac{1000}{1015}$$

Does this look right? There seems to be a lot more protons to neutrons?

 

[TFM]

Okay, so:

$$e^{2.07 * 10^{-13}/1.38 * 10^{-12} = 0.150}$$

So:

$$P(N) = \frac{e^{E(s_1)/k_BT}}{e^{E(s_1)/k_BT} + e^{E(s_2)/k_BT}} = \frac{0.15}{0.15 + 1} = \frac{0.15}{1.15} = \frac{15}{115}$$

and:

$$P(P) = \frac{e^{E(s_2)/k_BT}}{e^{E(s_1)/k_BT} + e^{E(s_2)/k_BT}} = \frac{1}{0.15 + 1} = \frac{1}{1.15} = \frac{100}{115}$$

Better?

 

[Vuldoraq]

Okay, so:

$$e^{2.07 * 10^{-13}/1.38 * 10^{-12} = 0.150}$$

Surely the exponential power shuold be negative here? And the above is wrong even for the values you have given, it should be equal to 1.1618, not 0.15? At the risk of being rude I would say bin your calculator, it is dragging you down!

 

[TFM]

Technically, it is excel, not a calculator...

Although, it sometimes requires the operator to remember to actually do the exponential, not just the division.

So:

$$e^{-E/k_BT} = 0.86$$

so:

P(P) = 1/(1+0.86) = 1/1.86 = 100/186

P(N) = 86/186

Is this better now?

 

[Vuldoraq]

Technically, it is excel, not a calculator...

Although, it sometimes requires the operator to remember to actually do the exponential, not just the division.

So:

$$e^{-E/k_BT} = 0.86$$

so:

P(P) = 1/(1+0.86) = 1/1.86 = 100/186

P(N) = 86/186

Is this better now?

Perfect! Excel sucks! Matlab is much better, you don't have to remember silly things like = (although it does have other annoyances).

Best

 

[TFM]

Excellent, Thanks

TFM