To determine if [tex]\( X \)[/tex] is a pure substance or a mixture and to calculate the mass of Element [tex]\( E_1 \)[/tex] in a new 10.0 g sample of [tex]\( X \)[/tex], let's go through the given data and perform the necessary analysis step by step.
### Step 1: Calculate the Ratio of [tex]\( E_1 \)[/tex] to [tex]\( E_2 \)[/tex] for Each Sample
We have four samples with their respective masses of [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] given:
1. Sample 1: [tex]\( E_1 = 6.3 \)[/tex] g, [tex]\( E_2 = 21.6 \)[/tex] g
2. Sample 2: [tex]\( E_1 = 4.1 \)[/tex] g, [tex]\( E_2 = 13.9 \)[/tex] g
3. Sample 3: [tex]\( E_1 = 6.8 \)[/tex] g, [tex]\( E_2 = 23.1 \)[/tex] g
4. Sample 4: [tex]\( E_1 = 5.9 \)[/tex] g, [tex]\( E_2 = 20.1 \)[/tex] g
The ratio for each sample can be calculated as follows:
[tex]\[ \text{Ratio}_1 = \frac{6.3}{21.6} \approx 0.2917 \][/tex]
[tex]\[ \text{Ratio}_2 = \frac{4.1}{13.9} \approx 0.2950 \][/tex]
[tex]\[ \text{Ratio}_3 = \frac{6.8}{23.1} \approx 0.2944 \][/tex]
[tex]\[ \text{Ratio}_4 = \frac{5.9}{20.1} \approx 0.2935 \][/tex]
### Step 2: Determine if [tex]\( X \)[/tex] is a Pure Substance
To determine if [tex]\( X \)[/tex] is a pure substance, we check if the ratio of [tex]\( E_1 \)[/tex] to [tex]\( E_2 \)[/tex] is consistent across all samples. As we see:
- [tex]\(\text{Ratio}_1 \approx 0.2917\)[/tex]
- [tex]\(\text{Ratio}_2 \approx 0.2950\)[/tex]
- [tex]\(\text{Ratio}_3 \approx 0.2944\)[/tex]
- [tex]\(\text{Ratio}_4 \approx 0.2935\)[/tex]
The ratios are not identical, but they are very close to each other. Given these ratios, it can be concluded that [tex]\( X \)[/tex] is highly likely to be a pure substance because these small variations may be due to rounding or experimental error.
Thus, [tex]\( X \)[/tex] is a pure substance.
### Step 3: Calculate the Mass of [tex]\( E_1 \)[/tex] in a New 10.0 g Sample of [tex]\( X \)[/tex]
If [tex]\( X \)[/tex] is a pure substance, the average ratio can be used to determine the mass of [tex]\( E_1 \)[/tex] in a new sample. First, let's find the average ratio:
[tex]\[ \text{Average Ratio} = \frac{0.2917 + 0.2950 + 0.2944 + 0.2935}{4} \approx 0.2937 \][/tex]
Using this average ratio, we can now calculate the mass of [tex]\( E_1 \)[/tex] in a 10.0 g sample of [tex]\( X \)[/tex].
Given the total mass of the sample [tex]\( X \)[/tex] is 10.0 g, let the mass of [tex]\( E_1 \)[/tex] be [tex]\( m \)[/tex] and the mass of [tex]\( E_2 \)[/tex] be [tex]\( 10.0 - m \)[/tex]. According to the average ratio:
[tex]\[ \frac{m}{10.0 - m} = 0.2937 \][/tex]
Solving for [tex]\( m \)[/tex]:
[tex]\[ m = 0.2937 \times (10.0 - m) \][/tex]
[tex]\[ m = 2.937 - 0.2937m \][/tex]
[tex]\[ m + 0.2937m = 2.937 \][/tex]
[tex]\[ 1.2937m = 2.937 \][/tex]
[tex]\[ m \approx \frac{2.937}{1.2937} \approx 2.27 \text{ g} \][/tex]
Therefore, the mass of Element [tex]\( E_1 \)[/tex] in a new 10.0 g sample of [tex]\( X \)[/tex] is approximately:
[tex]\[ \boxed{2.27 \text{ g}} \][/tex]
