Question

Boyle’s law If you have taken a chemistry class, then you are probably familiar with Boyle’s law: for gas in a confined space kept at a constant temperature, pressure times volume is a constant (in symbols, $PV=k$). Students collected the following data on pressure and volume using a syringe and a pressure probe.

(a) Make a reasonably accurate scatterplot of the data by hand using volume as the explanatory variable.

Describe what you see.

(b) If the true relationship between the pressure and volume of the gas is $PV=k$, we can divide both sides of this equation by $V$to obtain the theoretical model $P=k/V$, or $P=k(1/V)$. Use the graph below to identify the transformation that was used to linearize the curved pattern in part (a).

(c) Use the graph below to identify the transformation that was used to linearize the curved pattern in part (a).

Short answer

a). Direction: Negative because the scatterplot slopes upwards.

Form: Curved, because the points do not lie in a straight line.

Strength: Strong, because all points lie very close together in the same pattern.

b). The used transformation is the reciprocal of the volume.

c). The used transformation is the reciprocal of the pressure.

Step-by-step solution

Part (a) Step 1: Given Information

Part (a) Step 2: Explanation

The scatterplot of the following data given in the question taking volume as the explanatory variable is as follows:

Thus, by looking at the scatterplot we can say that,

Direction: Negative, because the scatterplot slopes upwards.

Form: Curved, because the points do not lie in a straight line.

Strength: Strong, because all points lie very close together in the same pattern.

Thus, this is the conclusion.

Part (b) Step 3: Given Information

Part (b) Step 4: Explanation

The plot is given in the question for part (b) and also it is given the two options:

$P=k\times \frac{1}{V}$

$\text{Or,}\frac{1}{P}=\frac{1}{k}\times V$

Thus, we note that the values on the horizontal axis are different from the value in the table for volume while the values on the vertical axis are the same values in the table for pressure. This then implies that the explanatory variable on the horizontal axis is the reciprocal of the volume and the response variable on the vertical axis is the pressure.

Part (c) Step 5: Given Information

Part (c) Step 6: Explanation

The plot is given in the question for part (c) and also it is given the two options:

$P=k\times \frac{1}{V}$

$\text{Or,}\frac{1}{P}=\frac{1}{k}\times V$

Thus, we note that the values on the horizontal axis are the same as the values for the volume while the values on the vertical axis are higher than the values for the pressure given in the table. This then means that the explanatory variable on the horizontal axis is the volume and the response variable on the vertical axis is the reciprocal of the pressure. This then implies that the used transformation is the reciprocal of the pressure.