At a temperature of 20°C, the volume V (in liters) of 1.33 g of O2 is related to its pressure p (in atmospheres) by the formula
V(p) = 1/p
(a) What is the average rate of change of V with respect to p as p increases from p = 7 to p = 8?
L/atm
(b) What is the rate of change to V with respect to p when p = 7?
L/atm
To find the average rate of change of V with respect to p, we'll need to determine the change in V divided by the change in p.
(a) Average rate of change of V with respect to p from p = 7 to p = 8:
V(p) = 1/p
To find the change in V, we substitute the values of p into the formula and calculate the difference:
V(8) - V(7) = 1/8 - 1/7
= (7 - 8)/(8 * 7)
= -1/56
To find the change in p, we subtract the initial value from the final value:
p = 8 - 7
= 1
Now, we divide the change in V by the change in p to find the average rate of change:
Average rate of change = (Change in V) / (Change in p)
= (-1/56) / (1)
= -1/56
Therefore, the average rate of change of V with respect to p as p increases from p = 7 to p = 8 is -1/56 L/atm.
(b) To find the rate of change of V with respect to p when p = 7, we need to take the derivative of the formula V(p) = 1/p with respect to p.
dV/dp = -1/p^2
Substituting p = 7 into the derivative:
dV/dp | p=7 = -1/(7^2)
= -1/49
Therefore, the rate of change of V with respect to p when p = 7 is -1/49 L/atm.
a) 7.5
b)2