Dec
21
I am taking physical chemistry and we are using partial derivatives, I have never solved partials and need?
Filed Under Physical Chemistry
a little help.The problem reads:
p=RT/V + (a+bT)/V^2
a and b are constants. Find (dV/dT) holding p constant.
dV/dT is a partial derivative.
It is a physical chemistry problem where we are looking at volume with respect to temperature while holding pressure constant. I attempted to isolate v, however I could not. Explanation would be helpful because I need to understand the process so I can repeat it in the future. Thanks in advance.
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take d /dT of teh enitre equation, remember p is held constant
0 = R/V - RT/V^2 dV/dT +b/V^2 - 2(a+bT)/V^3 dV/dT
Multiply by V^3
0 = RV^2 - RTV dV/dT +bV - 2(a+bT) dV/dT
0 = RV^2 + bV - (RTV + 2*[a+bT]) dV/dT
dV/dT = {RV^2 + bV}/(RTV + 2*[a+bT])
To find a partial derivative, for example, with respect to T, consider all other variables to be held constant. You do not need to isolate V. So the partial of V is
0 = RN + b/V^2 + -2(a + bT)(dV/dT)/V^3. Now you can solve for dV/dT.
It is like derivatizing by using every term as a variable (except for the constants which have a derivative of 0, and p which is held constant). So you get :
dp = 0 = RdT/V - (1/2)RTdV/V^2 + bdT/V^2 - (1/3)(a+bT)dV/V^3
by dividing by dT you get R/V - ((1/2)RT/V^2)dV/dT - b/V^2 -((1/3)(a+bT)/V^3)dV/dT =
Rassemble all the term and you will find dV/dT expressed as a differential equation.