#### To determine

**To express:** The volume of the gas as a function of the pressure of the gas.

#### Answer

The volume *V* of the gas is 5.3P_.

#### Explanation

**Given:**

The volume of the gas is V=0.106 m3.

The pressure of the gas is P=50 kPa.

**Boyle’s Law:**

“The pressure of the gas is inversely proportional to the volume of the gas.”

That is, P∝1V or k=PV, (1)

Where, *k* is the proportional constant.

**Calculation:**

Obtain the volume of the gas as a function of the pressure.

Substitute 0.106 for *V* and 50 for *P* in equation (1),

k=PV=50(0.106)=5.3

Thus, the value of k=5.3.

Substitute 5.3 for *k* in the equation (1),

PV=5.3V=5.3P

Therefore, the volume of the gas is V=5.3P.

#### To determine

**To calculate:** The derivative of *V* when P=50 kPa and to explain the meaning of the derivative and its units.

#### Answer

The value of dVdP|P=50=−0.00212.

#### Explanation

From part (a), the volume of the gas V=5.3P.

**Derivative rules:**

(1) Power Rule:ddx(xn)=nxn−1

(2) Constant Multiple Rule: ddx(cf(x))=cddx(f(x))

**Calculation:**

The derivative of *V* is dVdP, which is obtained as follows,

dVdP=ddP(5.3P)=ddP(5.3P−1)

Apply the constant multiple rule (2),

dVdP=5.3ddP(P−1)

Apply the power rule (1) and simplify the terms,

dVdP=5.3(−1P−1−1)=(5.3)(−1)P−2=−5.3P2

Therefore, the derivative of *V* is −5.3P2_.

The derivative of *V* at P=50 kPa

dVdP|P=50=−5.3502=−5.32500=−0.00212 m3kPa

The derivative dVdP means the rate of change of the volume with respect to pressure at 250C and its units is m3kPa_.