Derive lowest order (linear) approximation

[Jen2114]

Homework Statement

For a single mechanical unit lung, assume that the relationship among pressure, volume, and number of moles of ideal gas in the ling is given by P_A((V_L)/(N_L)^a = K, where a = 1 and K is a constant. Derive the lowest-order (linear approximation to the relationship among changes in pressure, changes in volume, and changes in moles of gas within the lung.

Homework Equations

The Attempt at a Solution

I know that to solve this you need to take the total derivative. However, I am having trouble arriving at the correct answer because I'm not really sure what to do with the derivative. So my calculations are the ones below:
P_A ((V_L)/(N_L)^a = K --> dPA/PA(PA(VL/NL)^a ) = dPA/PA(K) --> dPA/PA(VL/NL)^a-1 = 1

 

[tnich]

Homework Statement

For a single mechanical unit lung, assume that the relationship among pressure, volume, and number of moles of ideal gas in the ling is given by P_A((V_L)/(N_L)^a = K, where a = 1 and K is a constant. Derive the lowest-order (linear approximation to the relationship among changes in pressure, changes in volume, and changes in moles of gas within the lung.

Homework Equations

The Attempt at a Solution

I know that to solve this you need to take the total derivative. However, I am having trouble arriving at the correct answer because I'm not really sure what to do with the derivative. So my calculations are the ones below:
P_A ((V_L)/(N_L)^a = K --> dPA/PA(PA(VL/NL)^a ) = dPA/PA(K) --> dPA/PA(VL/NL)^a-1 = 1

I think you might find this problem easier if you took the log of both sides of the equation first before taking the derivative.

 

[Jen2114]

I think you might find this problem easier if you took the log of both sides of the equation first before taking the derivative.

Thanks, so doing that I got the following:
log(P_A(V_L/N_L)^)a = log(K) --> a*log(P_AV_L/N_L) ^{= log (K) --> a[log(P_A) + log(V_L) - log(N_L)] = log(K) using log properties for multiplication and division. Is this good so far?}

 

[tnich]

Thanks, so doing that I got the following:
log(P_A(V_L/N_L)^)a = log(K) --> a*log(P_AV_L/N_L) ^{= log (K) --> a[log(P_A) + log(V_L) - log(N_L)] = log(K) using log properties for multiplication and division. Is this good so far?}

That depends on what you mean by PA((VL)/(NL)^a = K in the problem statement. Is that $(\frac{P_AV_L}{N_L})^a = K$, $P_A(\frac{V_L}{N_L})^a = K$, or something else?

 

[Jen2114]

That depends on what you mean by PA((VL)/(NL)^a = K in the problem statement. Is that $(\frac{P_AV_L}{N_L})^a = K$, $P_A(\frac{V_L}{N_L})^a = K$, or something else?

It should be the second one where only V_L and N_L are raised to exponent variable a. So, in that case is
the result a[log(P_A) + log(V_L) - log(N_L) = log (K) correct?

 

[tnich]

It should be the second one where only V_L and N_L are raised to exponent variable a. So, in that case is
the result a[log(P_A) + log(V_L) - log(N_L) = log (K) correct?

No. Try doing it step-by-step.

 

[Jen2114]

No. Try doing it step-by-step.

So these are the steps I took
1. Divide both sides by P_A --> P_A(V_L/N_L)^a = K -->
(V_L/N_L)^a = K/P_A
2. Take log of both sides so a in exponent can be brought down --> log ( (V_L/N_L)^a )= log(K/P_A) --> a*log(V_L/N_L) = log((K/P_A))
3. Using log properties rewrite V_L/N_L and K/P_A as differences to get -->
log(V_L) - log(N_L) = 1/a *(log(K) - log(P_A) ... not sure what I am doing wrong so far

 

[Chestermiller]

Are you familiar with the product rule and the quotient rule for differentiation?

 

[tnich]

So these are the steps I took
1. Divide both sides by P_A --> P_A(V_L/N_L)^a = K -->
(V_L/N_L)^a = K/P_A
2. Take log of both sides so a in exponent can be brought down --> log ( (V_L/N_L)^a )= log(K/P_A) --> a*log(V_L/N_L) = log((K/P_A))
3. Using log properties rewrite V_L/N_L and K/P_A as differences to get -->
log(V_L) - log(N_L) = 1/a *(log(K) - log(P_A) ... not sure what I am doing wrong so far

As you did in the problem statement, you have left out the closing parenthesis in step 3. That notation error has apparently prompted you to misinterpret which terms are multiplied by 1/a.

 

[Jen2114]

As you did in the problem statement, you have left out the closing parenthesis in step 3. That notation error has apparently prompted you to misinterpret which terms are multiplied by 1/a.

Yes, sorry so what I mean is log(K) and log(P_A) are both multiplied by 1/a

 

[Jen2114]

Are you familiar with the product rule and the quotient rule for differentiation?

Yes, but I am not sure how it works for a total derivative of an expression.

 

[tnich]

Yes, sorry so what I mean is log(K) and log(P_A) are both multiplied by 1/a

Right. Now differentiate your equation.

 

[Jen2114]

Right. Now differentiate your equation.

Ok so then -->1. d/V_L(log(V_L)) - d/N_L(log(N_L)) = 1/a*(d/K(log(K))) - 1/a*(d/P_A(log(P_A)))
2. dV_L/V_L - dN_L/N_L = -1/a*dP_L/P_L and since K is a constant the derivative is 0 so it this correct? Thanks.

 

[Chestermiller]

Yes, but I am not sure how it works for a total derivative of an expression.

$$d\left[P\left(\frac{V_L}{N_L}\right)^a\right]=\left(\frac{V_L}{N_L}\right)^adP+aP\left(\frac{V_L}{N_L}\right)^{a-1}d\left(\frac{V_L}{N_L}\right)$$
$$d\left(\frac{V_L}{N_L}\right)=\frac{N_LdV_L-V_LdN_L}{N_L^2}$$

 

[Jen2114]

$$d\left[P\left(\frac{V_L}{N_L}\right)^a\right]=\left(\frac{V_L}{N_L}\right)^adP+aP\left(\frac{V_L}{N_L}\right)^{a-1}d\left(\frac{V_L}{N_L}\right)$$
$$d\left(\frac{V_L}{N_L}\right)=\frac{N_LdV_L-V_LdN_L}{N_L^2}$$

Oh ok this makes sense to me according to the product and quotient rule but the final solution given to us in class should be
dP_A/aP_A = -dV_L/V_L + dN_L/N_L so maybe I just can't see it but I'm not sure if the result you showed me simplifies to this so I know what the solution should be but I'm not sure how to arrive to that final solution.

 

[Chestermiller]

Oh ok this makes sense to me according to the product and quotient rule but the final solution given to us in class should be
dP_A/aP_A = -dV_L/V_L + dN_L/N_L so maybe I just can't see it but I'm not sure if the result you showed me simplifies to this so I know what the solution should be but I'm not sure how to arrive to that final solution.

It does reduct to that final solution. Just work on it a little. This is just algebra.

 

[tnich]

Ok so then -->1. d/V_L(log(V_L)) - d/N_L(log(N_L)) = 1/a*(d/K(log(K))) - 1/a*(d/P_A(log(P_A)))
2. dV_L/V_L - dN_L/N_L = -1/a*dP_L/P_L and since K is a constant the derivative is 0 so it this correct? Thanks.

Yes. That looks right.

 

[Jen2114]

It does reduct to that final solution. Just work on it a little. This is just algebra.

Ok, thanks! will do.

 

[Jen2114]

Yes. That looks right.

Thank you! , this really helped.

 

[tnich]

Thank you! , this really helped.

You are welcome. You are not quite done, yet. You still need to state your equation in terms of ΔP_A, ΔN_L, and ΔV_L. I am assuming your teacher has demonstrated something similar and you not required to give a formal proof. Otherwise, you would need to write out a Taylor series and do some more work.