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 P = kV^n to a linear form, we need to manipulate the equation to obtain a linear relationship between P and V.
Taking the logarithm of both sides of the equation gives:
log(P) = log(kV^n)
Now, we can use properties of logarithms to simplify the equation further. The logarithm of a product is equal to the sum of the logarithms of the individual factors, and the logarithm of a power is equal to the product of the power and the logarithm of the base.
Using these properties, we can rewrite the equation as:
log(P) = log(k) + n*log(V)
Let's define new variables for simplicity:
x = log(V)
y = log(P)
a = log(k)
Now, our equation becomes:
y = a + nx
We can see that the equation has a linear form, y = mx + c, where m is the slope (n) and c is the y-intercept (a).
To solve for the values of k and n using the given data points (110 cm^3, 50.3 cm) and (230 cm^3, 18.6 cm), we need to find the slope (n) and y-intercept (a).
1. For the first data point (110 cm^3, 50.3 cm):
Since we know that V = 110 cm^3 and P = 50.3 cm, we can take the logarithm of both values:
x₁ = log(110) and y₁ = log(50.3)
2. For the second data point (230 cm^3, 18.6 cm):
Similarly, x₂ = log(230) and y₂ = log(18.6)
Now, we have two sets of coordinates (x₁, y₁) and (x₂, y₂) that we can use to calculate the slope (n) and y-intercept (a) using the formula:
n = (y₂ - y₁) / (x₂ - x₁)
a = y₁ - n*x₁
Substituting the values, we get:
n = (log(18.6) - log(50.3)) / (log(230) - log(110))
a = log(50.3) - n*log(110)
Calculate these values to determine the slope (n) and y-intercept (a). Then, you can substitute the values back into the original equation to find the values of k.