For a particular mass of gas, the volume (Vcm^3) and pressure (Pcm) of mecury (Hg) are related by P=kV^n, where k and n are constants. Reduce the relation to a linear one and determine the values of k and n given that when v=110cm^3, P=50.3cm and when V=230cm^3, P=18.6cm
To reduce the relation to a linear one, we need to manipulate the given equation.
Starting with the equation P = kV^n, we can take the logarithm of both sides to make the equation logarithmic:
log(P) = log(kV^n)
Using logarithmic rules, we can break down the equation further:
log(P) = log(k) + log(V^n)
Next, we can use another logarithmic property to bring down the exponent:
log(P) = log(k) + n * log(V)
Now, let's define some new variables for simplicity:
Let x = log(P)
Let a = log(V)
Let b = log(k)
The equation now simplifies to:
x = b + n * a
We can form a linear equation using the given data points.
When V = 110 cm^3, P = 50.3 cm:
x₁ = log(50.3)
a₁ = log(110)
Similarly, when V = 230 cm^3, P = 18.6 cm:
x₂ = log(18.6)
a₂ = log(230)
Now we have two data points in the form (a, x):
(a₁, x₁) = (log(110), log(50.3))
(a₂, x₂) = (log(230), log(18.6))
We can use these data points to find the slope (n) and y-intercept (b) of the linear equation, which will give us the values of k and n.
First, find the slope (n):
n = (x₂ - x₁) / (a₂ - a₁)
n = (log(18.6) - log(50.3)) / (log(230) - log(110))
Calculate n using the above formula.
Next, find the y-intercept (b):
b = x₁ - n * a₁
b = log(50.3) - n * log(110)
Calculate b using the above formula.
Once you have calculated n and b, substitute these values back into the original equation to determine the values of k and n:
P = kV^n
Finally, you will have determined the values of k and n for the given relationship.