ADIABATIC EXPANSION
when a certain polyatomic gas undergoes adiabatic expansion,its pressure p and volume V satisfy the equation pV^1.3 =k,where k is constant.Find the relationship between the related rates dp/dt and dV/dt.
this is a problem of related rates please show all solutions and diagram thanks
p v^1.3 = k
p d/dt (v^1.3) + v^1.3 dp/dt = 0
1.3 p v^.3 dv/dt + v^1.3 dp/dt = 0
1.3 p dv/dt = - v dp/dt
1.3 dv/dt /v = - dp/dt /p
thanks =) we have arrived at the same answer
Oh, related rates! I guess we're getting into some gas business now. Alright, let's put our thinking caps on and solve this puzzle!
Given that we got an adiabatic expansion, we know that there's no heat exchange happening here. So, we can assume that it's an isentropic process. Fun, isn't it?
Now, we're given the equation pV^1.3 = k, where k is a constant. We need to find the relationship between the related rates dp/dt and dV/dt.
First, let's take the derivative of the equation with respect to time:
dp/dt * V^1.3 + p * 1.3V^0.3 * dV/dt = 0
Now, let's solve it for dp/dt:
dp/dt = -p * 1.3V^0.3 * dV/dt / V^1.3
Simplifying this further, we get:
dp/dt = -1.3p * dV/dt / V
And there you have it! The relationship between the related rates dp/dt and dV/dt is given by:
dp/dt = -1.3p * dV/dt / V
I hope this helps! Just remember, in the world of adiabatic expansions, no heat means no stress.
To find the relationship between the related rates dp/dt and dV/dt, we will differentiate the given equation with respect to time (t).
Given: pV^1.3 = k
Differentiating both sides with respect to time (t):
d/dt (pV^1.3) = d/dt (k)
Using the product rule for differentiation, we have:
dp/dt * V^1.3 + p * d/dt (V^1.3) = 0
Next, we need to find dV/dt, which represents the rate of change of the volume V with respect to time t.
Differentiating the equation pV^1.3 = k with respect to time (t), we have:
d/dt (pV^1.3) = d/dt (k)
Using the chain rule, we have:
dp/dt * V^1.3 + p * dV^1.3/dt = 0
Using the power rule for differentiation, we have:
dp/dt * V^1.3 + p * d/dt (V^1.3) = 0
dp/dt * V^1.3 + p * 1.3 * V^(1.3-1) * dV/dt = 0
dp/dt * V^1.3 + 1.3pV^0.3 * dV/dt = 0
Comparing the equations for dp/dt * V^1.3 + p * dV^1.3/dt = 0 and dp/dt * V^1.3 + 1.3pV^0.3 * dV/dt = 0, we can see that the two equations are equal.
Therefore, the relationship between the related rates dp/dt and dV/dt is:
dp/dt = - (1.3pV^0.3 / V^1.3) * dV/dt
or equivalently:
dp/dt = - (1.3p / V) * dV/dt
This is the relationship between the rates of change of pressure and volume in an adiabatic expansion of a polyatomic gas.
To find the relationship between the related rates dp/dt and dV/dt, we need to differentiate the given equation with respect to time (t) using the chain rule.
Differentiating pV^1.3 = k with respect to t, we get:
d/dt (pV^1.3) = d/dt (k)
By applying the chain rule, we have:
dp/dt * V^1.3 + p * d/dt (V^1.3) = 0
Next, we need to find d/dt (V^1.3). To do this, we differentiate V^1.3 with respect to t:
d/dt (V^1.3) = dV/dt * 1.3V^0.3
Now substitute this result back into our previous equation:
dp/dt * V^1.3 + p * (dV/dt * 1.3V^0.3) = 0
Simplifying this equation, we get:
dp/dt * V^1.3 + 1.3pV^0.3 * dV/dt = 0
Finally, we can rearrange the equation to isolate the related rates:
dp/dt = (-1.3pV^0.3 * dV/dt) / V^1.3
dp/dt = (-1.3p * dV/dt) / V
Therefore, the relationship between the related rates dp/dt and dV/dt is given by:
dp/dt = (-1.3p * dV/dt) / V
Note: It is not possible to provide a diagram for this specific problem as it involves a mathematical derivation rather than a physical representation.