The maximum ratio of PEP to pyruvate is obtained when
, so we want
The activity of phosphate is irrelevant (at least, in this problem) so we have
This is not a very large ratio, but it is considerably better than what would be obtained without ATP and may be sufficient if only a small amount of phosphoenolpyruvate is needed.
- There are at least two different approaches to this
problem:
- Method 1
- Boiling has to do with the process
. We begin by
computing the equilibrium constant at 298K:
At the boiling point, K=1. (Make sure you understand why that is before proceeding.) We use the formula which relates equilibrium constants to temperature, which we rearrange to the form
- Method 2
- At the normal boiling point, liquid water is
in equilibrium with 1atm of steam thus K=1
for the process .
This implies that
. Therefore
.
(This is Trouton's rule.)
Since we can calculate 
(as
above) and 
, we can compute
the temperature T at which
, i.e. the boiling
point:
- Both of these methods underestimate the boiling point by
for essentially the same reason:
Both assume that is independent of
temperature, which is not quite right. The second
method also assumes that is
independent of temperature.
and 
, so
. Since H=E+PV,
Therefore
.
Note: If I put a differential derivation on a test or exam, it will be an optional question. If you learn how to do these, they are quite easy. However, make sure that you can at least interpret a differential (i.e. relate the differential to derivatives of state functions) that is given to you.
- Imagine taking a sample of water at 298K and warming or cooling
it reversibly to temperature T. Adding the change in
entropy to the entropy at the starting temperature, we
get
where T is in Kelvin and the entropy is in
. - Take a Taylor series near
:
where
. - Here are some entropies calculated with the two equations:
The approximation is valid to 1% for
up to
about 80K, although the approximation is better on the
high-temperature side than on the low-T side.
. (This is somewhat arbitrary and you could
of course pick a different number.)
To obtain a conservative estimate of the maximum ionic strength
for ideal behaviour, take the largest reasonable values for
and 
: There are few anions stable in solution whose
charges are other than -1 or -2 so take 
. There are
some +3 and +4 cations so take 
. The critical ionic
strength is therefore
Below an ionic strength of 
, we can be
reasonably assured of ideal behaviour. We also know that above
0.01mol/L, the Debye-Hückel equation yields poor results so
we should use Debye-Hückel theory when
.
or
Since the charges on the calcium and oxalate ions are +2 and
-2, 
. Accordingly
Start iteration with the guesstimate 
:
The molar mass of calcium oxalate is 128.10g/mol so the
solubility is 
.
Note that the ionic strength of the saturated solution is
, which is well within the
range of applicability of Debye-Hückel theory.
. Using the freezing-point depression formula,
the total concentration of solutes should therefore
be
Since NaCl dissociates into 
and 
,
this is twice the concentration of either sodium or chloride
ions (i.e. twice the formal concentration of sodium chloride).
The formal concentration of sodium chloride should therefore be
1.4mol/kg.
We have 18000kg of solvent so
In the more conventional concentration units, this is
. Since 100mL of water
(0.100L) of water was used, the number of moles of the
forensic sample is 
.
The mass was 2.841mg so the molar mass of the unknown
substance is
From the chemical formula, we know that the molar mass of MDA is 179.218g/mol. It seems quite likely that the material brought in by the police officer is MDA. Further tests on the sample should be ordered.