Let's go through each part of the question step-by-step:
### 1. Calculate [tex]\(P \times V\)[/tex]:
We are given pressures and volumes as follows:
- 1 atm and 8 L
- 2 atm and [tex]\(a\)[/tex] L (where [tex]\(a\)[/tex] is unknown)
- 4 atm and 2 L
- 8 atm and 1 L
We need to calculate the product [tex]\(P \times V\)[/tex] for each pair of [tex]\(P\)[/tex] and [tex]\(V\)[/tex].
- For the first pair: [tex]\(1 \text{ atm} \times 8 \text{ L} = 8 \text{ atm·L}\)[/tex]
- For the second pair: [tex]\(2 \text{ atm} \times a \text{ L} = 2a \text{ atm·L}\)[/tex]
- For the third pair: [tex]\(4 \text{ atm} \times 2 \text{ L} = 8 \text{ atm·L}\)[/tex]
- For the fourth pair: [tex]\(8 \text{ atm} \times 1 \text{ L} = 8 \text{ atm·L}\)[/tex]
Thus, the PV values for the known pressures and volumes are:
- [tex]\(8 \text{ atm·L}\)[/tex]
- [tex]\(2a \text{ atm·L}\)[/tex] (unknown value)
- [tex]\(8 \text{ atm·L}\)[/tex]
- [tex]\(8 \text{ atm·L}\)[/tex]
### 2. What is the peculiarity in the value of PV?
The peculiarity is that the product [tex]\(P \times V\)[/tex] remains constant for the known values:
- [tex]\(8 \text{ atm·L}\)[/tex]
- [tex]\(8 \text{ atm·L}\)[/tex]
- [tex]\(8 \text{ atm·L}\)[/tex]
This indicates that the product [tex]\(P \times V\)[/tex] does not change, which is a characteristic feature of a specific gas law.
### 3. Identify the gas law related to this:
The constancy of the product [tex]\(P \times V\)[/tex] is indicative of Boyle's Law. According to Boyle's Law, for a given amount of gas at constant temperature, the pressure and volume are inversely proportional to each other, meaning [tex]\(P \times V\)[/tex] remains constant.
### 4. Write down the mathematical expression that represents this gas law:
The mathematical expression for Boyle's Law is:
[tex]\[ P_1 \times V_1 = P_2 \times V_2 \][/tex]
### 5. What will be the volume of this gas at 16 atm pressure?
Given the initial pressure and volume:
[tex]\[ P_1 = 8 \text{ atm}, \; V_1 = 1 \text{ L}\][/tex]
And the new pressure:
[tex]\[ P_2 = 16 \text{ atm} \][/tex]
We use Boyle's Law to find the new volume [tex]\(V_2\)[/tex]:
[tex]\[ P_1 \times V_1 = P_2 \times V_2 \][/tex]
[tex]\[ 8 \text{ atm} \times 1 \text{ L} = 16 \text{ atm} \times V_2 \][/tex]
[tex]\[ V_2 = \frac{8 \text{ atm} \times 1 \text{ L}}{16 \text{ atm}} \][/tex]
[tex]\[ V_2 = \frac{8}{16} \text{ L} \][/tex]
[tex]\[ V_2 = 0.5 \text{ L} \][/tex]
Hence, the volume of the gas at 16 atm pressure will be [tex]\(0.5 \text{ L}\)[/tex].
In summary:
1. [tex]\(P \times V = 8 \text{ atm·L}\)[/tex], [tex]\(2a \text{ atm·L}\)[/tex], [tex]\(8 \text{ atm·L}\)[/tex], [tex]\(8 \text{ atm·L}\)[/tex]
2. The peculiarity is that [tex]\(P \times V\)[/tex] remains constant for the known values.
3. This is Boyle's Law.
4. The mathematical expression is [tex]\(P_1 \times V_1 = P_2 \times V_2\)[/tex].
5. The volume at 16 atm pressure is [tex]\(0.5 \text{ L}\)[/tex].
