If TV^k-1 = constant
Show PV^k=constant
And P^1-k T^k =constant.
k=(cp/cv)
To show that PV^k is constant given TV^(k-1) = constant, we can use the ideal gas law equation: PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
Let's start by expressing the given relationship in terms of pressure and volume:
TV^(k-1) = constant ...........(1)
Rearranging equation (1), we have:
V^(k-1) = constant/T ...........(2)
Now, let's express the ideal gas law equation in terms of pressure and volume:
PV = nRT ...........(3)
Solving equation (3) for P, we get:
P = nRT/V ...........(4)
Next, substitute equation (4) into equation (2):
(nRT/V)^(k-1) = constant/T ...........(5)
Multiply both sides of equation (5) by V^(k-1):
(nRT)^k = constantV^(k-1) ...........(6)
Now, we can see that (nRT)^k is equal to PV^k, and constantV^(k-1) simplifies to a new constant:
PV^k = constant ...........(7)
Therefore, PV^k is constant when TV^(k-1) is constant.
Moving on to the second statement "P^(1-k)T^k = constant," we can derive it using the given expression for k, which is k = (cp/cv), where cp and cv represent specific heat capacities.
Rearranging the equation, we have:
cv = cp/k ............(8)
Multiplying both sides by k:
k*cv = cp ............(9)
Now, let's use the ideal gas law equation and equation (9) to express the relationship between pressure, volume, and temperature:
PV = nRT
PV = (cp/k)nRT .........using equation (9)
Rearranging the equation, we have:
P^(1-k)T^k = (cp/k)nR * V^(1) * V^(-k) ............(10)
Extracting the term (cp/k)nR as a constant, we get:
P^(1-k)T^k = constant * V^(1) * V^(-k) ............(11)
Simplifying the right-hand side of equation (11), we have:
P^(1-k)T^k = constant * V^(1-k) ............(12)
Since (1-k) is equal to -k, equation (12) becomes:
P^(1-k)T^k = constant * V^(-k) ............(13)
Now, we know from equation (2) that V^(k-1) = constant/T. If we raise both sides to the power of -1, we get:
V^(-k) = T/constant ............(14)
Substituting equation (14) into equation (13), we have:
P^(1-k)T^k = constant * (T/constant) ............(15)
Simplifying equation (15), we get:
P^(1-k)T^k = T ............(16)
Finally, we can see that P^(1-k)T^k is equal to T, which means P^(1-k)T^k is a constant.
Therefore, we have shown that PV^k = constant and P^(1-k)T^k = constant, where k = (cp/cv).