A substance has an isothermal compressibility kappa = (aT^3)/(P^2)...

In summary, the conversation discusses the implementation of conditions when T and P are constant, resulting in the equations ln V = aT^3/P + constant and ln V = bT^3 /3P + constant. The assumption that the constant is 0 leads to the conclusion that a/b = 1/3. To justify this assumption, the speaker proceeds to differentiate and proves that b/3 -a must be zero, making f a function of T and P. The initial conditions of V_0, P_0, and T_0 are used to find the constant in the equation of state, ln V = bT^3 / 3P + ln V_0 - b(T_0 )^3
  • #1
romanski007
12
1
Homework Statement
A substance has isothermal compressibility kappa = (aT^3)/(P^2) and an expansivity beta = (bT^2)/P where a and b are constants.
i) find the equations of state of the substance and the ratio a/b.
Relevant Equations
kappa = (aT^3)/(P^2)
beta = (bT^2)/P
Starting from v(P,T),
dv=(dv/dp)_T dp + (dv/dT)_P dt
i implemented conditions when T and P are constant and ended up with
ln V = aT^3/P + constant and ln V = bT^3 /3P + constant

If i assume that the constant is 0, i can say that a/b = 1/3 but how do i justify this assumption?
 
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  • #2
I get, after one integration ##\ln{V}=\frac{bT^3}{3P}+f(P)##
 
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  • #3
I proceeded, then differentiated wrt P and got that f'(P) = (T^3/P^2)(b/3 -a ).
Hence I proved that b/3 -a must be zero by contradiction as otherwise f would be a function of T and P.
Hence a/b = 1/3, and f'(T) = 0 so that f'(T) is come constant.
Initial conditions would be V_0 P_0 and T_0. hence ln V_0 = b(T_0 )^3/3P_0 + const and const can be found.
equation of state would be ln V = bT^3 / 3P + ln V_0 - b(T_0 )^3/3P_0
Is this correct? Thanks.
 
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Related to A substance has an isothermal compressibility kappa = (aT^3)/(P^2)...

1. What does isothermal compressibility mean?

The isothermal compressibility of a substance is a measure of how much the substance's volume changes in response to a change in pressure at a constant temperature.

2. How is the isothermal compressibility calculated?

The isothermal compressibility (kappa) is calculated by dividing the change in volume (V) by the change in pressure (P) at a constant temperature (T). It is often expressed in units of inverse pressure (1/P).

3. What does the equation kappa = (aT^3)/(P^2) represent?

This equation represents the relationship between isothermal compressibility (kappa), temperature (T), and pressure (P). The constant "a" is specific to the substance and can be used to compare the compressibility of different substances.

4. How does temperature affect isothermal compressibility?

In general, as temperature increases, the isothermal compressibility of a substance also increases. This means that the substance becomes more compressible at higher temperatures.

5. What is the significance of knowing the isothermal compressibility of a substance?

The isothermal compressibility can provide important information about the behavior of a substance under different pressures and temperatures. It is often used in thermodynamics and fluid mechanics to understand the properties of gases and liquids.

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