Hello,
The pressure P and volume V of an expanding gas are related by the formula PVb=C, where b and C are constants (this holds in adiabatic expansion, without heat gain or loss).
Find dPdt if b=1.6, P=12kPa, V=80cm2, and dVdt=20cm3/min.
Thus far,
I got d(PV)/dt= p(dv/dt) + V (dp/dt), which can be rewritten to dp/dt = -P/V * (dv/dt)
However, when I plug the numbers in, they are incorrect!
-12/80 * 20 = -3
What am I doing wrong?
Thank you in advance.
Well, it seems like you've made a mathematical error in your calculation. Let's go through it again step by step.
You correctly started with the equation dp/dt = -P/V * (dv/dt).
Now, let's plug in the given values: P = 12 kPa, V = 80 cm^2, dv/dt = 20 cm^3/min.
Now, let's convert the units so they match. Since 1 cm^2 = 10^-4 m^2 and 1 L = 1000 cm^3, we get P = 12 * 10^3 Pa and dv/dt = 20 * 10^-6 m^3/s.
Plugging these values in, we have dp/dt = - (12 * 10^3 Pa) / (80 * 10^-4 m^3) * (20 * 10^-6 m^3/s).
Simplifying, we have dp/dt = -0.015 m^2/s^2.
So, the correct value for dP/dt is -0.015 m^2/s^2.
I hope this helps and apologies for any confusion. Keep up the good work!
Your calculations are indeed incorrect. Let's go through the steps to find the correct answer.
You correctly started with the equation PV^b = C, where b is a constant. Considering that we need to find dP/dt, we can differentiate both sides of the equation with respect to time.
Differentiating PV^b = C with respect to time (t), we get:
d(PV^b)/dt = dC/dt
Using the product rule of differentiation, the left side becomes:
P(dV^b/dt) + V^b(dP/dt) = 0
Since we are given that dV/dt = 20 cm^3/min, we can substitute this value into the equation:
P(dV^b/dt) + V^b(dP/dt) = 0
P(bV^(b-1)) (dV/dt) + V^b(dP/dt) = 0
Plugging in the values b = 1.6, P = 12 kPa, V = 80 cm^2, and dV/dt = 20 cm^3/min:
12(1.6)(80^(0.6))(20) + (80^1.6)(dP/dt) = 0
To find dP/dt, rearrange the equation:
(80^1.6)(dP/dt) = -12(1.6)(80^(0.6))(20)
dP/dt = [-12(1.6)(80^(0.6))(20)] / (80^1.6)
Now you can plug in the values into the equation to find dP/dt.
From the information you provided, you correctly manipulated the equation PVb=C to obtain dp/dt = -P/V * (dV/dt). However, it seems that you made an error when plugging in the numbers.
Let's go through the calculations step by step:
1. Start with the given equation: PVb = C.
2. Take the derivative of both sides with respect to time (t):
d/dt (PVb) = d/dt (C).
3. Apply the product rule on the left side of the equation:
P(bVb-1)(dV/dt) + bVb(dP/dt) = 0.
4. Rearrange the equation to isolate (dP/dt):
dP/dt = -P(b/dV)(dV/dt).
5. Substitute the given values:
P = 12 kPa,
V = 80 cm³,
dV/dt = 20 cm³/min,
b = 1.6.
Now, let's plug these values in:
dP/dt = -(12 kPa)(1.6/80 cm³)(20 cm³/min).
Converting cm³ to m³, we have:
dP/dt = -(12,000 Pa)(0.02 m³/80 m³)(20 m³/min).
Simplifying further:
dP/dt = -(12,000)(0.02)(20) Pa/min.
dP/dt = -4800 Pa/min.
Therefore, the correct value for dP/dt is -4800 Pa/min, not -3.
I think the error may have been in the unit conversion from cm³ to m³. Make sure to convert all the units to be consistent for accurate calculations.
You want dP/dt, you have dV/dt so what do you need?
dP/dt = dV/dt * something
the something is dP/dV so that the dVs cancel and so you get dP on top.
You have to write PVb=C as P = and then differentiate that to get dP/dV.
You should be able to substitute everything in then after that
You can find what C is from the start by substituting in b, V and P