A physical chemist happened to know that the
𝑏
value for a gas that followed the Van der Waals equation was
𝑏
=
0.05
L
/
m
o
l
. But he didn't know the
𝑎
value. He got 1.00 mole of the gas at a pressure of 1.00 atm at 10.0 °C and found that it had a volume of 23.0 L. Based on this single measurement, what is the probable value of
𝑎
? Note that
(
𝑝
+
𝑎
𝑛
2
𝑉
2
)
(
𝑉
−
𝑛
𝑏
)
=
𝑛
𝑅
𝑇
is the Van der Waals equation of state,
𝑅
=
0.08206
L
a
t
m
m
o
l
−
1
K
−
1

Question

A physical chemist happened to know that the 
𝑏
 value for a gas that followed the Van der Waals equation was 
𝑏
=
0.05
L
/
m
o
l
.  But he didn't know the 
𝑎
 value.  He got 1.00 mole of the gas at a pressure of 1.00 atm at 10.0 °C and found that it had a volume of 23.0 L.  Based on this single measurement, what is the probable value of 
𝑎
?  Note that
(
𝑝
+
𝑎
𝑛
2
𝑉
2
)
(
𝑉
−
𝑛
𝑏
)
=
𝑛
𝑅
𝑇
 is the Van der Waals equation of state, 
𝑅
=
0.08206
L
a
t
m
m
o
l
−
1
K
−
1

BlackTom AI · Accepted Answer

The problem gives b = 0.05 L/mol for a gas described by the Van der Waals equation. We’re told: n = 1 mole, P = 1.00 atm, T = 10.0 °C (which is 283.15 K), V = 23.0 L. The Van der Waals equation is (P + a n^2 / V^2)(V − nb) = nRT. With n = 1, this becomes (P + a / V^2)(V − b) = RT.

First, compute V − b: V − nb = 23.0 − 0.05 = 22.95 L.
Next, compute P + a / V^2: V^2 = 23.0^2 = 529 L^2, so P + a / V^2 = 1.00 + a/529.
Then RT for the given T is R × T with R = 0.08206 L·atm/(mol·K) and T = 283.15 K, so RT ≈ 0.08206 × 283.15 ≈ 23.23 L·atm/mol.

Plug into the equation: (1 + a/529) × 22.95 ≈ 23.23.
From there, 1 + a/529 ≈ 23.23 / 22.95 ≈ 1.0124.
Thus a/529 ≈ 0.0124, giving a ≈ 529 × 0.0124 ≈ 6.56.
Rounding to one decimal, a ≈ 6.6 (units: L^2·atm/mol^2).

Now evaluate each option:
- Option 6.6: This matches the calculated value within rounding, so it is consistent with the data and the Van der Waals relation.
- Option -510: A large negative value for a would imply a negative attraction term a n^2 / V^2 that would drastically distort P + a/V^2. Given the measured P, V, and T, this would not satisfy the equation, so this is implausible.
- Option -6.6: A small negative a would yield P + a/V^2 < P, reducing the left-hand side. However, using the given numbers, the left-hand side must be just slightly above RT/V−b to satisfy the equation; a negative a of this magnitude would not reconcile the equation with the RT value, making this incorrect.
- Option none of the others: Since a ≈ 6.6 provides a consistent solution, claiming none of the others would be incorrect.
- Option 510: A very large positive a would push P + a/V^2 up substantially, which would then require a much larger (V − b) or RT to balance, conflicting with the fixed RT value. This makes 510 inconsistent with the given data.

In summary, the computation shows a ≈ 6.6 L^2·atm/mol^2 best fits the provided measurements under the Van der Waals equation, while the other choices fail to satisfy the equation with the given P, V, and T.

类似问题

There are two adjustable parameters, 𝑎 and 𝑏 that appear in the Van der Waals gas equation of state. Based on what you understand their roles to be, how would they compare between C F 4 and C F 2 C l 2 ? (The carbon atom is central in each compound.)

Which of the following describes the van der Waals equation of state?

When evaluating a new project, firms should include in the projected cash flows all of the following EXCEPT:

Provide the best rationale for what's wrong with supply chains today?

What is Blockchain and how is it relevant to supply chain management?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单