Question

activity c:
the ideal gas law
get the gizmo ready:

• select volume as the dependent variable.

introduction: so far you have explored the individual relationships between p, v, t, and n. in this activity, you will combine all of these relationships into a single law, enabling you to see how the behavior of a gas is affected when several different variables are changed at one time.
question: how do volume, pressure, temperature, and amount of gas interact?

1. analyze: avogadro’s law states that the volume of a gas is directly proportional to the number of moles (n). if two variables are directly proportional the symbol “∝” is used, which means “directly proportional to.” therefore, the relationship between volume and moles would be written as v ∝ n. this type of relationship is known as a proportionality.

a. volume is also directly proportional to temperature (t). express this relationship as a proportionality, in the following form: v ∝ ______
b. since volume is inversely proportional to pressure, volume is directly proportional to 1/p. express this relationship as a proportionality: _________

1. synthesize: if you know that a ∝ b and a ∝ c, then you can also say that a ∝ bc, or the product of b and c. take the above three proportionalities (including v ∝ n) and combine them into a single proportionality in the form: v ∝? show your work below.
2. calculate: a proportionality is not the same thing as an equation, however. to convert a proportionality into an equation it is necessary to multiply by a mathematical constant. for example, a ∝ b means that a = kb, where k is a constant. when referring to gases, this constant is referred to as r, the ideal gas constant.

a. in the space at right, rewrite the proportionality you created in question 2 so that the proportionality symbol (∝) is changed to “=” and the right side is multiplied by r.
b. rearrange your equation to solve for the ideal gas constant: r =
c. you should have gotten an equation equivalent to (r = pv / nt). pick any set of conditions in the gizmo you would like, and then calculate r. show your work below.
p = ____ v = ____
n = ____ t = __ r = ____
d. recalculate r using a completely different set of gizmo conditions: r = ______
(activity c continued on next page)

Explanation:

Step 1: Analyze Part A

The problem states volume is directly proportional to temperature (\(T\)). Direct proportionality is denoted by \(\propto\), so the relationship is \(V \propto T\).

Step 2: Analyze Part B

Volume is inversely proportional to pressure (\(P\)), meaning it's directly proportional to \(1/P\). So the proportionality is \(V \propto \frac{1}{P}\).

Step 3: Synthesize the Proportionalities

We have three proportionalities: \(V \propto n\) (from Avogadro's law), \(V \propto T\) (from part A), and \(V \propto \frac{1}{P}\) (from part B). When combining proportionalities, if \(a \propto b\), \(a \propto c\), and \(a \propto d\), then \(a \propto bcd\). So combining these, \(V \propto \frac{nT}{P}\).

Step 4: Convert Proportionality to Equation (Part A)

To convert \(V \propto \frac{nT}{P}\) to an equation, we multiply by the ideal gas constant \(R\). So \(V = R \cdot \frac{nT}{P}\) or \(V = \frac{nRT}{P}\).

Step 5: Solve for \(R\) (Part B)

Starting from \(V = \frac{nRT}{P}\), we can rearrange to solve for \(R\). Multiply both sides by \(P\): \(PV = nRT\). Then divide both sides by \(nT\): \(R = \frac{PV}{nT}\).

Step 6: Calculate \(R\) (Part C and D)

For example, let's pick conditions (hypothetical Gizmo values, say standard temperature and pressure for simplicity, though Gizmo may have its own values). Let \(P = 1\) atm, \(V = 22.4\) L, \(n = 1\) mol, \(T = 273.15\) K. Then \(R = \frac{PV}{nT} = \frac{1 \cdot 22.4}{1 \cdot 273.15} \approx 0.0821\) L·atm/(mol·K). For a different set, say \(P = 2\) atm, \(V = 11.2\) L, \(n = 1\) mol, \(T = 273.15\) K. Then \(R = \frac{2 \cdot 11.2}{1 \cdot 273.15} \approx 0.0821\) L·atm/(mol·K) (showing \(R\) is constant under ideal conditions).

Answer:

s:

1. A. \(T\) (so \(V \propto T\))

B. \(V \propto \frac{1}{P}\)

1. \(V \propto \frac{nT}{P}\)
2. A. \(V = \frac{nRT}{P}\)

B. \(R = \frac{PV}{nT}\)
C. (Example) \(P = 1\) atm, \(V = 22.4\) L, \(n = 1\) mol, \(T = 273.15\) K, \(R = \frac{1 \times 22.4}{1 \times 273.15} \approx 0.0821\) L·atm/(mol·K)
D. (Example) \(P = 2\) atm, \(V = 11.2\) L, \(n = 1\) mol, \(T = 273.15\) K, \(R = \frac{2 \times 11.2}{1 \times 273.15} \approx 0.0821\) L·atm/(mol·K)