Question

consider the graph shown below, then select all of the statements that are false.
i. at a given temperature, curve a represents the gas particles with the lowest molecular mass.
ii. for a given gas sample, curve d represents the highest temperature.
iii. at a given temperature, curve d represents gas particles with the highest molar mass.
iv. at a given temperature, the gas particles represented by curve c are slightly lighter then those represented by curve b.
options:
iv only
i,ii,iii,iv
ii,iv
ii only

Explanation:

Step1: Analyze Statement I

For a given temperature, lighter gas particles (lower molar mass) have higher average molecular speeds. Curve A has the highest molecular speed, so it should represent the lowest molar mass (lightest particles). So Statement I is **True** (not false).

Step2: Analyze Statement II

Higher temperature increases the average molecular speed and broadens the distribution. Curve D has the lowest molecular speed (narrowest, most peaked at low speed), so it represents the lowest temperature, not the highest. So Statement II is **False**.

Step3: Analyze Statement III

At a given temperature, higher molar mass means lower average speed. Curve D has the lowest speed, so it should represent the highest molar mass (heaviest particles). So Statement III is **True** (not false).

Step4: Analyze Statement IV

Lighter particles have higher speed. Curve C has a higher speed than Curve B? Wait, no—wait, Curve B and C: if C is slightly lighter than B, C should have a higher speed. But looking at the graph, Curve B and C: wait, actually, for same temperature, lighter mass → higher speed. If C is lighter than B, C's peak should be at higher speed. But in the graph, Curve B and C: let's see, Curve C is to the right of B? Wait, no, the x - axis is molecular speed. Wait, Curve A is rightmost (highest speed), then B, C, D? Wait, no, the graph: Curve A is the rightmost (highest speed), then B, C, D (leftmost). Wait, no, the curves: A is the one with the peak at highest speed (right), D at lowest (left). So for B and C: if C is slightly lighter than B, C should have a higher speed (peak at higher x - value) than B. But in the graph, is C to the right of B? Wait, the curves: B and C—looking at the graph, Curve C is to the right of B? Wait, no, the original graph: let's see, the curves are A (rightmost), then B, C, D (leftmost). Wait, no, the labels: D is the rightmost? Wait, no, the y - axis is relative number of particles, x - axis is molecular speed (0 to 2500). So Curve A: starts at 0, rises, then falls, peak at highest speed (right). Curve D: peak at lowest speed (left). So between B and C: Curve C is to the right of B? Wait, no, maybe I got it reversed. Wait, the problem says "the gas particles represented by curve C are slightly lighter than those represented by curve B". If C is lighter, C should have higher speed (peak at higher x). So if Curve C is to the right of B, then C has higher speed, so C is lighter. But in the graph, let's check: if B and C are two curves, and C is slightly to the right of B, then C has higher speed, so C is lighter. But wait, the statement is "slightly lighter then those represented by curve B"—wait, maybe a typo, "than". So if C is lighter than B, C should have higher speed. So is Curve C to the right of B? If yes, then C is lighter, so the statement is True? Wait, no—wait, the question is which statements are FALSE. Wait, maybe I made a mistake. Wait, let's re - evaluate.

Wait, the key for molecular speed: \( v_{rms}=\sqrt{\frac{3RT}{M}} \), so \( v \propto \frac{1}{\sqrt{M}} \) (at constant T). So lower M → higher v.

So for Statement IV: "At a given temperature, the gas particles represented by curve C are slightly lighter than those represented by curve B." So if C is lighter than B, \( M_C < M_B \), so \( v_{rms,C}>v_{rms,B} \). So Curve C should be to the right of Curve B (higher speed). Looking at the graph, is Curve C to the right of Curve B? If yes, then the statement is True. But wait, maybe the curves are such that B is to the right of C? Wait, the original graph: the curves are A (rightmost), then B, C, D (leftmost)? Or D, C, B, A? Wait, the labels: D is the top - right curve? Wait, no, the graph shows: the y - axis is relative number of particles, x - axis is molecular speed (0 - 2500). The curves start at (0,0), rise to a peak, then fall. Curve A has the peak at the highest x (rightmost), Curve D at the lowest x (leftmost). Then between them: B and C. Let's assume the order from left (low speed) to right (high speed) is D, C, B, A. So D (leftmost, low speed), then C, then B, then A (rightmost, high speed). So if C is to the left of B, then \( v_{rms,C}<v_{rms,B} \), so \( M_C > M_B \) (since \( v \propto 1/\sqrt{M} \)). So if the statement says C is lighter than B ( \( M_C < M_B \) ), but in reality, if C is to the left of B (lower speed), \( M_C > M_B \), so the statement is False? Wait, maybe I misread the graph. Alternatively, maybe the curves are A (right), B, C, D (left). So A (highest speed), B, C, D (lowest). So B is to the right of C, so \( v_B > v_C \), so \( M_B < M_C \) (since \( v \propto 1/\sqrt{M} \) ). So if the statement says C is slightly lighter than B ( \( M_C < M_B \) ), but \( M_C > M_B \), so the statement is False. Wait, this is confusing. But let's go back.

Wait, the question is to find which statements are FALSE. We know Statement II is False. Now, let's re - check:

Statement I: At a given temperature, curve A represents the gas particles with the lowest molecular mass. Since \( v \propto 1/\sqrt{M} \), lowest M → highest v. Curve A has highest v, so I is True.

Statement II: For a given gas sample, curve D represents the highest temperature. Higher T → higher v and broader distribution. Curve D has lowest v (narrowest, most peaked at low v), so it's lowest T. So II is False.

Statement III: At a given temperature, curve D represents gas particles with the highest molar mass. Highest M → lowest v. Curve D has lowest v, so III is True.

Statement IV: At a given temperature, the gas particles represented by curve C are slightly lighter than those represented by curve B. If C is lighter than B, \( M_C < M_B \) → \( v_C > v_B \). So Curve C should be to the right of B. If in the graph, B is to the right of C ( \( v_B > v_C \) ), then \( M_B < M_C \) (since \( v \propto 1/\sqrt{M} \) ), so C is heavier than B, so the statement is False. Wait, but maybe the graph shows C to the right of B? Let's assume the graph: Curve A (rightmost), then B, then C, then D (leftmost). So A (highest v), B, C, D (lowest v). So \( v_A > v_B > v_C > v_D \), so \( M_A < M_B < M_C < M_D \). Then Statement IV: C is slightly lighter than B. But \( M_C > M_B \), so C is heavier than B. So the statement is False. Wait, but earlier I thought III was True. Wait, no—III: "At a given temperature, curve D represents gas particles with the highest molar mass." Since \( M_D > M_C > M_B > M_A \), yes, D has highest M. So III is True.

Wait, but now we have II and IV as False? But the options are:

Options:

- IV only

- I,II,III,IV

- II,IV

- II only

Wait, maybe my analysis of IV is wrong. Let's re - consider the formula \( v_{rms}=\sqrt{\frac{3RT}{M}} \). So for two gases at the same T, \( \frac{v_{rms,1}}{v_{rms,2}}=\sqrt{\frac{M_2}{M_1}} \). So if \( v_C \approx v_B \) (slightly different), then \( M_C \approx M_B \), and if \( v_C > v_B \), \( M_C < M_B \); if \( v_C < v_B \), \( M_C > M_B \).

Looking at the graph, Curve B and Curve C: let's say Curve B is to the right of Curve C (so \( v_B > v_C \) ), then \( M_B < M_C \) (since \( v \propto 1/\sqrt{M} \) ). So the statement "the gas particles represented by curve C are slightly lighter than those represented by curve B" means \( M_C < M_B \), but we have \( M_C > M_B \), so the statement is False.

Statement II: "For a given gas sample, curve D represents the highest temperature." For a given gas (fixed M), higher T → higher \( v_{rms} \) and broader distribution. Curve D has the lowest \( v_{rms} \) (narrowest distribution), so it's the lowest temperature, not highest. So II is False.

Statement I: True, Statement III: True, Statements II and IV: False. Wait, but the options: one of the options is "II,IV". Let's check the options again:

Options:

- IV only

- I,II,III,IV

- II,IV

- II only

So if II and IV are False, then the correct option is "II,IV".

Answer:

II,IV