Robert Boyle (1627–1691) found that for a given quantity of ga...

Question

Robert Boyle (1627–1691) found that for a given quantity of gas at a constant temperature, the pressure P (in kPa) and volume V of the gas (in m³) are accurately approximated by the equation V=k/P, where k>0 is constant. Suppose the volume of an expanding gas is increasing at a rate of 0.15 m³/min when the volume V=0.5 m³ and the pressure is P=50 kPa. At what rate is pressure changing at this moment?

Accepted Answer

Welcome back, everyone. In this problem, Avogajo's law states that for a given temperature and pressure, the volume V in liters and the number of moles n of a gas are directly proportional by the given equation V equals KN, where K greater than 0 is a constant. Suppose the volume of a gas is increasing at a rate of 0.3 L per minute when the volume V equals 5 L and the number of moles is N equals 1 mole. Determine the rate at which the number of moles is changing at this moment. A is 0.06 moles per minute. B 0.06 moles per minute, C 0.03 moles per minute, and D, negative 0.03 moles per minute. Now how are we going to determine the rate at which the number of moles is changing? Let's think about the information that we already have, OK? So far, we know that the rate at which the volume is changing, DVDT, is equal to 0.3 L per minute, OK? And that's the case when the volume is 5 L and the number of moles N equals 1 mole. How are we going to use this to help us figure out DNDT, OK? DNDT, that is how the moles are changing the rate at which the mo the number of moles are changing. Well, do we know anything that relates to the number of moles and the volume? Well, we know Avogaja's law. Recall, like it tells us, the volume and the number of moles of a gas are directly proportional, and we're given the equation V E equals kn. So if we differentiate both sides with respect to the time, we'll get the derivative of the volume with respect to time, the derivative of the number of moles with respect to time, and we should be able to substitute to rearrange our equation to solve for the rate of change for the number of moles. So differentiating. With respect to tea. OK, then. We're going to get DZDT to be equal to K multiplied by DNDT. Now we know DVDT, is equal to 0.3 L per minute. OK. So, 0.3 L per minute equals K multiplied by the NDT, OK? And no. That means then that the NDT is gonna be equal to 0.3 L per minute. Divided by K. Now can we figure out what K is? Remember K is a constant, and we can figure out K using V and N. Remember, we're already told that at this rate, our volume is 5 L and N equals 1 mole. So K equals 5 L divided by 1 mole or 5 L per mole. So since we have a value of K, then that means the rate of change of the number of moles will be equal to 0.3 L per minute, divided by. 0, sorry, 5 L per mole. Which gives us a rate of 0.06 moles per minute. Thus, 0.06 moles per minute is the rate at which the number of moles is changing at this moment. Therefore, A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Related Rates

Implicit Differentiation

Boyle's Law

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