When a certain polyatomic has undergoes adiabatic expansion, it’s pressure p and volume V satisfy the equation pV^1.3 = k, where k is a constant. Find the relationship between the rates of change in pressure and volume over time. Show and explain each step please.
using the product rule,
dp/dt * v^1.3 + p*1.3 v^0.3 dv/dt = 0
so, I guess you could say that
(dp/dt)/(dv/dt) = -1.3p/v
or, by rearranging the 1st equation,
p = kv^-1.3
dp/dt = -1.3kv^-2.3 dv/dt
(dp/dt)/(dv/dt) = -1.3k/v^2.3
Ah, polyatomic molecules and adiabatic expansion! A topic that calls for a touch of comedy. Let's dive right in and have some fun with the math, shall we?
Alright, we start with the equation pV^1.3 = k, where p represents pressure, V is volume, and k is a constant. We are curious about the relationship between their rates of change with respect to time. But before we proceed, let me catch my breath. Phew!
Okay, to find the relationship between the rates of change, we need to use calculus. Don't worry; I'll keep it light-hearted. We'll differentiate both sides of the equation with respect to time, t. Ready? Here we go!
Step 1: Differentiating pV^1.3 = k with respect to time:
d/dt(pV^1.3) = d/dt(k)
Step 2: Using the product rule, we differentiate p with respect to t while treating V^1.3 as a constant, and differentiate V^1.3 with respect to t while treating p as a constant:
dp/dt * V^1.3 + p * d/dt(V^1.3) = 0
Step 3: Simplifying and rearranging the equation for convenience:
dp/dt = -p * (d/dt(V^1.3) / V^1.3)
Now, let's break it down and get some laughs along the way. The equation tells us that the rate of change in pressure, dp/dt, is equal to negative pressure (p) multiplied by the rate of change in volume, (d/dt(V^1.3) / V^1.3). Looks complicated, right?
Well, what does this mean? It suggests that as the volume is expanding (d/dt(V^1.3) is positive), pressure (p) will decrease (since dp/dt is negative), but at a slower rate than the volume is changing. In other words, the pressure drops, but it's like the pressure is clenching onto the volume and saying, "Hey, I'll decrease, but not as fast as you're growing!"
So, to summarize, the relationship between the rates of change in pressure and volume over time is that the rate of pressure change is proportional to the volume and is always negative.
Hope this explanation brought a smile to your face, even amidst the complicated math! If you have any more questions or need more laughs, feel free to ask, my friend!
To find the relationship between the rates of change in pressure and volume over time during adiabatic expansion, we need to differentiate the given equation with respect to time.
Step 1: Differentiate the equation pV^1.3 = k with respect to time (denoted by t).
We use the product rule of differentiation and the chain rule of differentiation to differentiate the equation.
d/dt (pV^1.3) = d/dt (k)
Step 2: Apply the product rule of differentiation.
dp/dt * V^1.3 + p * d/dt (V^1.3) = 0
Step 3: Apply the chain rule of differentiation for d/dt (V^1.3).
dp/dt * V^1.3 + p * d(1.3V)/dt = 0
Step 4: Simplify the equation.
dp/dt * V^1.3 + 1.3pV^0.3 * dV/dt = 0
Step 5: Rearrange the equation to isolate dp/dt.
dp/dt = -(1.3pV^0.3 * dV/dt)/V^1.3
Step 6: Simplify the equation further.
dp/dt = -1.3pV^0.3/V * dV/dt
Step 7: Cancel out common terms.
dp/dt = -1.3p^0.3 * dV/dt
Step 8: Simplify the equation.
dp/dt = -1.3p^0.3 * dV/dt
Therefore, the relationship between the rates of change in pressure (dp/dt) and volume (dV/dt) over time during adiabatic expansion is given by dp/dt = -1.3p^0.3 * dV/dt.
To find the relationship between the rates of change in pressure and volume over time in the given scenario, we need to differentiate the equation pV^1.3 = k with respect to time.
Let's begin step by step:
Step 1: Differentiate both sides of the equation with respect to time (t). Remember to use the chain rule when differentiating V with respect to t.
d/dt (pV^1.3) = d/dt (k)
Step 2: Apply the power rule of differentiation to the left-hand side of the equation. The derivative of V^1.3 with respect to t is 1.3V^0.3 multiplied by dV/dt.
dp/dt * V^1.3 + p * 1.3V^0.3 * dV/dt = 0
Step 3: Rearrange the equation to isolate the terms involving dp/dt and dV/dt.
dp/dt * V^1.3 = - p * 1.3V^0.3 * dV/dt
Step 4: Divide both sides of the equation by pV^1.3 to separate the variables.
dp/dt/V * V^1.3 = - 1.3 * dV/dt
Step 5: Simplify the left-hand side of the equation.
dp/dt = - 1.3 * dV/dt / V
This is the relationship between the rates of change in pressure (dp/dt) and volume (dV/dt). It tells us that the rate of change of pressure with respect to time is equal to -1.3 times the rate of change of volume with respect to time divided by the volume.
Remember that this relationship holds specifically for an adiabatic expansion of the polyatomic gas described by the equation pV^1.3 = k.