Question

1. in figure 4 below, a graph shows the relationship between mass and volume for two substances, a and b. use the graph to answer questions about these two substances.

a) you have built a simple two - pan balance shown above to compare the masses of substances a and b. what would happen to the balance if you put equal masses of a and b in the two pans? equal volumes of a and b in the two pans? explain your reasoning.
b) find the slope of the line for both a and b using correct units. state the physical meaning of the slope for each substance.
c) if you put 10.0 ml of a in one balance pan, what mass of b would you need in the other pan to make it balance? explain your reasoning.
d) if you put 35.0 ml of b in one balance pan, what volume of a would you need in the other pan to make it balance? explain your reasoning.
e) water has a density of 1.00 g/ml. sketch the line representing water on the graph in figure 4.

Explanation:

Step1: Analyze balance with equal - mass

If equal masses of A and B are put in the two pans, the balance will be in equilibrium because mass is the same on both sides.

Step2: Analyze balance with equal - volume

From the graph, the slope of the mass - volume graph gives density ($
ho=\frac{m}{V}$). The density of A is greater than that of B. For equal volumes, the mass of A ($m_A=
ho_A V$) will be greater than the mass of B ($m_B=
ho_B V$), so the pan with A will go down.

Step3: Calculate slope for A

For substance A, choose two points on the line, say (0,0) and (20, 40). The slope $m_A=\frac{\Delta m}{\Delta V}=\frac{40 - 0}{20 - 0}=2$ g/mL. The physical meaning is the density of substance A.

Step4: Calculate slope for B

For substance B, choose two points, say (0,0) and (40, 20). The slope $m_B=\frac{\Delta m}{\Delta V}=\frac{20 - 0}{40 - 0}=0.5$ g/mL. The physical meaning is the density of substance B.

Step5: Solve for mass of B to balance A

The density of A is 2 g/mL. For 10.0 mL of A, $m_A=
ho_A V_A=2\times10 = 20$ g. Let the mass of B be $m_B$. Since $m_B=m_A$ for balance, and $
ho_B = 0.5$ g/mL, then $m_B = 20$ g.

Step6: Solve for volume of A to balance B

The density of B is 0.5 g/mL. For 35.0 mL of B, $m_B=
ho_B V_B=0.5\times35 = 17.5$ g. Let the volume of A be $V_A$. Since $m_A=m_B$ and $
ho_A = 2$ g/mL, then $V_A=\frac{m_A}{
ho_A}=\frac{17.5}{2}=8.75$ mL.

Step7: Sketch water line

The density of water is 1.00 g/mL. The line for water will pass through the origin (0,0) and another point such as (10,10) (since $m=
ho V$ and when $V = 10$ mL, $m = 10$ g). Draw a straight - line through these points on the graph.

Answer:

a) Equal masses: balance is in equilibrium; Equal volumes: pan with A goes down.
b) Slope of A: 2 g/mL (density of A); Slope of B: 0.5 g/mL (density of B).
c) 20 g of B.
d) 8.75 mL of A.
e) Draw a straight - line passing through (0,0) and (10,10) on the graph.