Solve 2 simultaneous equations with 2 unknowns

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lmarshall1995

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I have the two below simultaneous equations and want to rearrange to find n

(dε/dt)_σ1 = Bσ1^n

and

(dε/dt)_σ2 = Bσ2^n

Note, the values I have are below:
(dε/dt)_σ1 = 0.075
(dε/dt)_σ2 = 0.1
σ1 = 10.0
σ2 = 11.0

I know B is a constant but do not know what it is, but mainly want to find the value for n.

Is it something to do with logs?

I have tried log(0.075/0.10)^(10.0/11.0) but I'm not sure if that is the right approach.

Thank you
 
Yes, to solve for an unknown exponent, use the logarithm.
From \(\displaystyle \left(\frac{d\epsilon}{dt}\right)_{\sigma_1}= B\sigma_1^n\)
we get \(\displaystyle log\left(\left(\frac{d\epsilon}{dt}\right)_{\sigma_1}\right)= log B+ nlog(\sigma_1)\).
Using the values you give, that is log(0.075)= log(B)+ n log(10.0).

From \(\displaystyle \left(\frac{d\epsilon}{dt}\right)_{\sigma_2}= B\sigma_2^n\)
we get \(\displaystyle log\left(\left(\frac{d\epsilon}{dt}\right)_{\sigma_2}\right)= log B+ nlog(\sigma_2)\).
Using the values you give, that is log(0.1)= log(B)+n log(11.0).

Now, subtracting one equation from the other eliminates B!
log(0.1)- log(0.075)= n(log(11.0)- log(10.0)

log(1/0.075)= n log(1.10)

log(13.333...)= n log(1.10)

n= log(13.333...)/log(1.10)
 
HallsofIvy said:
Yes, to solve for an unknown exponent, use the logarithm.
From \(\displaystyle \left(\frac{d\epsilon}{dt}\right)_{\sigma_1}= B\sigma_1^n\)
we get \(\displaystyle log\left(\left(\frac{d\epsilon}{dt}\right)_{\sigma_1}\right)= log B+ nlog(\sigma_1)\).
Using the values you give, that is log(0.075)= log(B)+ n log(10.0).

From \(\displaystyle \left(\frac{d\epsilon}{dt}\right)_{\sigma_2}= B\sigma_2^n\)
we get \(\displaystyle log\left(\left(\frac{d\epsilon}{dt}\right)_{\sigma_2}\right)= log B+ nlog(\sigma_2)\).
Using the values you give, that is log(0.1)= log(B)+n log(11.0).

Now, subtracting one equation from the other eliminates B!
log(0.1)- log(0.075)= n(log(11.0)- log(10.0)

log(1/0.075)= n log(1.10)

log(13.333...)= n log(1.10)

n= log(13.333...)/log(1.10)
Click to expand...
Thank you. This is very helpful and has confirmed my thought process. I appreciate the help. :)

Taking the same variables that one step further, if I have the below simultaneous equations, How would I find Q, then subsequently A?

(dε/dt)_T1 = Aσ^n exp[-Q/RT1]

and

(dε/dt)_T2 = Aσ^n exp[-Q/RT2]

Note, the values I have are below:
(dε/dt)_T1 = 0.075
(dε/dt)_T2 = 0.125
σ = 10.0
n = 4.0
R=8.314 J/K/mol
T1 = 290
T2 = 315

A is a constant (not area). I believe you have to find out Q first and then A?

Thank you again, I really appreciate your help.
 
Last edited:
Hi,

If I have the below simultaneous equations, How would I find Q and then A?

(dε/dt)T1 = Aσn exp[-Q/RT1]

and

(dε/dt)T2 = Aσn exp[-Q/RT2]

Note, the values I have are below:
(dε/dt)T1 = 0.075
(dε/dt)T2 = 0.125
σ = 10.0MN/m2
n = 4.0
R = 8.314 J/K/mol
T1 = 290K
T2 = 315K

A is a constant (not area). I believe you have to find out Q first and then A?

My thought process is to take log of both sides for both equations, and then subtract one from the other.

I get to the below but I'm not sure how to continue:

Log(dε/dt)T2 - Log(dε/dt)T1 = Log(exp[-Q/(RT2) - Log(exp[-Q/(RT1)

Then do I have to differentiate/integrate somehow to find Q?

Once I have the value of Q, I can rearrange the below for A

(dε/dt) = Aσn exp[-Q/RT

to become

A = (dε/dt) / σn exp[-Q/RT

and plug in the variables so I think I'l alright with that bit.

Thank you :)
 
lmarshall1995 said:
Hi,

If I have the below simultaneous equations, How would I find Q and then A?

(dε/dt)T1 = Aσn exp[-Q/RT1]

and

(dε/dt)T2 = Aσn exp[-Q/RT2]

Note, the values I have are below:
(dε/dt)T1 = 0.075
(dε/dt)T2 = 0.125
σ = 10.0MN/m2
n = 4.0
R = 8.314 J/K/mol
T1 = 290K
T2 = 315K

A is a constant (not area). I believe you have to find out Q first and then A?

My thought process is to take log of both sides for both equations, and then subtract one from the other.

I get to the below but I'm not sure how to continue:

Log(dε/dt)T2 - Log(dε/dt)T1 = Log(exp[-Q/(RT2) - Log(exp[-Q/(RT1)

Then do I have to differentiate/integrate somehow to find Q?

Once I have the value of Q, I can rearrange the below for A

(dε/dt) = Aσn exp[-Q/RT

to become

A = (dε/dt) / σn exp[-Q/RT

and plug in the variables so I think I'l alright with that bit.

Thank you :)
Click to expand...
Another way:

You have:

0.075 = Aσn exp[-Q/(8.314*290)] ......................................(1)

and

0.125 = Aσn exp[-Q/(8.314*315)] ......................................(2)

If divide (1) by (2) -what do you get?
 
Subhotosh Khan said:
Another way:

You have:

0.075 = Aσn exp[-Q/(8.314*290)] ......................................(1)

and

0.125 = Aσn exp[-Q/(8.314*315)] ......................................(2)

If divide (1) by (2) -what do you get?
Click to expand...

1.667 = e(0.000031701Q)

Then Q = 16120.2 ?

Then plugging in the variables into
A = (dε/dt) / σn exp[-Q/RT]

Would give me

A = 0.075 / [10.04.0 exp[-16120.2/8.314*290]]

A = 1.656492 × 10244193

That result seems rather big to me. Have I interpreted what you said correctly?- thank you for your help and patience.
 
Otis said:
Hello Imarshall. Double-check your arithmetic. I got different results for 0.075/0.125 and -0.0004147553358Q + 0.0003818382457Q.

\(\;\)
Click to expand...
Looks like I did (2)/(1) instead of (1)/(2)...
Thank you

Should have read:
0.6 = e^(-0.000031701 Q)

Then Q=16113.9

A = 0.075 / [(10.0^4.0)*exp[-16113.9/8.314*290]]

A=6.068136617206707 × 10^244097

That's an even bigger number! I must have something else wrong here. :-/
 
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