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.

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