Question

1. the 7th hr class produced the graph below when they plotted mass vs. volume for samples of two substances.

fig 4
80
70
substance a
60
50
mass (g)
40
30
20
substance b
10
0
10 20 30 40 50 60 70 80
volume (ml)

a. write the equation of the line for substance b.

b. how does substance b differ from substance a?

c. calculate the mass of a 35.0 cm³ piece of substance b.

d. what would occupy a larger volume: 100 g of substance a or 100 g of substance b? explain.

e. use a particle diagram to show the difference in density between the two substances.

Explanation:

Step1: Find the slope and y - intercept for substance B

The graph of substance B is a straight - line passing through the origin (0,0). The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. Taking two points on the line of substance B, say (0,0) and (80,40), the slope $m=\frac{40 - 0}{80 - 0}=\frac{1}{2}$. The equation of a line in slope - intercept form is $y = mx + b$, where $b$ is the y - intercept. Since the line passes through the origin, $b = 0$. So the equation for substance B is $y=\frac{1}{2}x$, or in terms of mass $m$ and volume $V$, $m=\frac{1}{2}V$.

Step2: Compare the lines of substance A and B

The slope of the line for substance A is steeper than that of substance B. The slope of a mass - volume graph represents density ($
ho=\frac{m}{V}$). So substance A has a higher density than substance B.

Step3: Calculate the mass of a 35.0 $cm^3$ piece of substance B

Using the equation $m=\frac{1}{2}V$, substitute $V = 35.0\ cm^3$. Then $m=\frac{1}{2}\times35.0\ g=17.5\ g$.

Step4: Determine which substance occupies a larger volume for 100 g

For substance A, let its density be $
ho_A$ and for substance B, $
ho_B$. Since $
ho_A>
ho_B$ (from the steeper slope of A's graph), and using the formula $V=\frac{m}{
ho}$, for a given mass $m = 100\ g$, the volume $V$ is inversely proportional to the density. So 100 g of substance B will occupy a larger volume than 100 g of substance A.

Step5: Particle diagram concept

In a particle diagram for density comparison, for substance A (higher density), the particles would be more closely packed together, while for substance B (lower density), the particles would be more spread out.

Answer:

a. $m=\frac{1}{2}V$
b. Substance A has a higher density than substance B (steeper slope of its mass - volume graph).
c. 17.5 g
d. 100 g of substance B would occupy a larger volume because substance B has a lower density than substance A.
e. In the particle diagram, particles of substance A are more closely packed, while particles of substance B are more spread out.