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
V(p) = 1/p
V(6) = 1/6
V(7) = 1/7
ΔV = V(7) - V(6) = 1/7 - 1/6 = -1/42
Δp = 7 - 6 = 1
a)Average rate of change:
ΔV/Δp = (-1/42)/1 = -1/42 Liter/atmosphere
Actual rate of change
dV/dp = -1/p²
At p = 6
dV/dp = -1/6² = -1/36 L/atm
(a) Well, let's see...If we've got the equation V(p) = 1/p, then the derivative of V with respect to p would be -1/p^2, right?
So, if we want to find the average rate of change of V with respect to p as p increases from 7 to 8, we can use the formula:
Average rate of change = (V(8) - V(7)) / (8 - 7)
Plugging in the values, we get:
Average rate of change = (1/8 - 1/7) / (8 - 7)
= (7 - 8) / (56 - 49)
= -1 / 7
So, the average rate of change of V with respect to p is -1/7 L/atm.
(b) Now, to find the rate of change of V with respect to p when p = 7, we can just use the derivative we found earlier:
Rate of change = -1 / 7^2
= -1 / 49
So, the rate of change of V with respect to p when p = 7 is -1/49 L/atm.
Hope that helps!
To find the average rate of change of V with respect to p, we need to calculate the difference in V divided by the difference in p.
(a) We are given that p increases from 7 to 8. Plugging these values into the formula V(p) = 1/p:
V(7) = 1/7
V(8) = 1/8
The difference in V is V(8) - V(7):
V(8) - V(7) = (1/8) - (1/7)
Now, we can calculate the average rate of change:
Average rate of change = (V(8) - V(7)) / (8 - 7)
= [(1/8) - (1/7)] / (8 - 7)
Simplifying this expression, we get:
Average rate of change = [(7 - 8) / (8 * 7)] / 1
= [-1 / 56] / 1
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 can use the derivative of the function V(p). The derivative gives us the instantaneous rate of change.
Differentiating V(p) = 1/p with respect to p, we get:
dV/dp = -1/p^2
Now, we can substitute p = 7 into the derivative:
dV/dp = -1/(7^2)
Simplifying the expression, we find:
dV/dp = -1/49 L/atm
Therefore, the rate of change of V with respect to p when p = 7 is -1/49 L/atm.
To find the average rate of change of V with respect to p as p increases from 7 to 8, we need to calculate the change in V divided by the change in p. Here's how to do it step by step:
(a) Average rate of change of V with respect to p:
Step 1: Calculate the initial volume V at p = 7.
Using the formula V(p) = 1/p, substitute 7 for p:
V(7) = 1/7
Step 2: Calculate the final volume V at p = 8.
Using the formula V(p) = 1/p, substitute 8 for p:
V(8) = 1/8
Step 3: Calculate the change in V.
Change in V = V(8) - V(7) = (1/8) - (1/7)
Step 4: Calculate the change in p.
Change in p = 8 - 7 = 1
Step 5: Calculate the average rate of change of V with respect to p.
Average rate of change = Change in V / Change in p
Average rate of change = [(1/8) - (1/7)] / 1
Simplifying the equation further:
Average rate of change = [7/56 - 8/56] / 1
Average rate of change = [-1/56] / 1
Average rate of change = -1/56
The units for V(p) are liters per atmosphere (L/atm). Therefore, the average rate of change of V with respect to p as p increases from 7 to 8 is -1/56 L/atm.
(b) Rate of change of V with respect to p when p = 7:
To find the rate of change of V at a specific value of p, we need to find the derivative of the function V(p) = 1/p. The derivative represents the rate of change of V with respect to p at any given point.
Using calculus, we can find the derivative of V(p) = 1/p:
Step 1: Take the derivative of V(p) with respect to p using the power rule for differentiation.
dV/dp = -1/p^2
Step 2: Substitute p = 7 into the derivative to find the rate of change at p = 7.
dV/dp = -1/(7^2) = -1/49
So, the rate of change of V with respect to p when p = 7 is -1/49 L/atm.
V(p) = 1/p
V(6) = 1/6
V(7) = 1/7
ΔV = V(7) - V(6) = 1/7 - 1/6 = -1/42
Δp = 7 - 6 = 1
a)Average rate of change:
ΔV/Δp = (-1/42)/1 = -1/42 Liter/atmosphere