In an adiabatic process (one in which no heat transfer takes place), the pressure [tex]$P$[/tex] and volume [tex]$V$[/tex] of an ideal gas such as oxygen satisfy the equation
[tex]$P^5 V^7=C$[/tex]
where C is a constant. Suppose that at a certain instant of time, the volume of the gas is 5 L, the pressure is 100 KPA, and the pressure is decreasing at the rate of 5 KPA/SEC. Find the rate at which the volume is changing.
Answer (1)
Differentiate the equation P 5 V 7 = C with respect to time t using implicit differentiation, obtaining 5 P 4 d t d P V 7 + 7 P 5 V 6 d t d V = 0 .
Solve the differentiated equation for d t d V , resulting in d t d V = − 7 P 5 V d t d P .
Substitute the given values V = 5 L, P = 100 kPa, and d t d P = − 5 kPa/sec into the equation.
Calculate the rate of change of volume: d t d V = 28 5 L/sec. The final answer is 28 5 L / SEC .
Explanation
Problem Setup We are given an adiabatic process where the pressure P and volume V of an ideal gas satisfy the equation P 5 V 7 = C , where C is a constant. We are also given that at a certain instant, the volume V = 5 L, the pressure P = 100 kPa, and the rate of change of pressure d t d P = − 5 kPa/sec. Our goal is to find the rate at which the volume is changing, which is d t d V .
Differentiating the Equation To find d t d V , we need to differentiate the given equation with respect to time t . Using implicit differentiation and the chain rule, we have: d t d ( P 5 V 7 ) = d t d ( C ) Applying the product rule, we get: 5 P 4 d t d P V 7 + P 5 ( 7 V 6 d t d V ) = 0
Solving for dV/dt Now, we solve for d t d V :
7 P 5 V 6 d t d V = − 5 P 4 V 7 d t d P d t d V = 7 P 5 V 6 − 5 P 4 V 7 d t d P d t d V = − 7 P 5 V d t d P
Substituting Values We are given V = 5 L, P = 100 kPa, and d t d P = − 5 kPa/sec. Substituting these values into the equation for d t d V , we get: d t d V = − 7 ( 100 ) 5 ( 5 ) ( − 5 ) d t d V = 700 25 × 5 d t d V = 700 125 d t d V = 28 5
Final Answer Therefore, the rate at which the volume is changing is 28 5 L/sec.
Examples
In thermodynamics, understanding adiabatic processes is crucial for designing engines and refrigerators. For instance, when compressing a gas in an engine cylinder very quickly, the process can be approximated as adiabatic. Knowing the relationship between pressure and volume, and how they change with time, allows engineers to optimize engine performance and efficiency. By applying calculus to these relationships, they can predict and control the changes in volume and pressure, leading to better designs.
