Chapter 3, Distribution of Molecular Weight Video Solutions, Physical Chemistry of Macromolecules: Basic Principles and Issues

Problem 1

Show that (a) the equation $\bar{r}_{\mathrm{n}}=\sum_1^{\infty} r p^{r-1}(1-p)$ leads to the equation $\overline{\boldsymbol{r}}_{\mathrm{n}}=1 /(1-p)$ and (b) the equation $\overline{\boldsymbol{r}}_{\mathrm{w}}=\sum_1^{\infty} r^2 p^{r-1}(1-p)^2$ leads to $r_{\mathrm{w}}=(1+p) /(1-p)$, where $\bar{r}_{\mathrm{n}}$ is the number average degree of polymerization and $\bar{r}_{\mathrm{w}}$ is the weight average degree of polymerization.

Victor Salazar

Numerade Educator

01:02

Problem 2

Show that $\bar{x}=m$ for the binomial, Poisson, and normal distributions.

Raj Bala

Numerade Educator

01:47

Problem 3

The most probable distribution function of the molecular weight of condensation polymers is given by
$$
w_r=r p^{r-1}(1-p)^2
$$
(a) Plot $w_r$ (in the range between 0 and 0.20 ) versus $r$ (in the range between 0 and 50 ) for $p=0.5,0.8,0.9$.
(b) Plot $w_{\mathrm{r}}\left(0 \leq w_r \leq .04\right)$ versus $r(0 \leq r \leq 250)$ for $p=0.9$ (Flory, 1936).

Crystal Wang

Numerade Educator

01:13

Problem 4

The Poisson distribution of the molecular weight of addition polymers (which do not have termination) is given by
$$
w_r=\frac{\gamma}{\gamma+1} e^{-\gamma} \frac{r \gamma^{r-2}}{r-1} !
$$

Plot $w_r$ in percentage $\left(0 \leq w_r \leq 6\right)$ versus $r(0 \leq r \leq 140)$ for $\gamma=50$, 100, 500 (Flory, 1940).

Josee Pacheco

Numerade Educator

03:46

Problem 5

Show that the mole fraction distribution $X_x$ is the same as the number average distribution $n_x$, that is, $X_x=n_x$.

Lottie Adams

Numerade Educator

02:56

Problem 6

Show that the number average molecular weight $M_{\mathrm{n}}$ is given by
$$
M_{\mathrm{n}}=\frac{\sum M_i N_i}{\sum N_i}=M_0 \sum_1^{\infty} X n_x=\frac{M_0}{1-p}
$$
and that the weight average molecular weight $M_{\mathrm{w}}$ is given by
$$
M_{\mathrm{w}}=\frac{\sum M_i^2 N_i}{\sum M_i N_i}=M_0 \sum X w_x=M_0 \frac{1+p}{1-p}
$$
where $M_0$ is the molecular weight of a monomer unit.

Ameer Said

Numerade Educator

02:26

Problem 7

Consider a solution containing equal numbers of molecules of molecular weights $50 \times 10^3, 100 \times 10^3, 200 \times 10^3$, and $400 \times 10^3$. Calculate $\bar{M}_{\mathrm{n}}, \overline{\boldsymbol{M}}_{\mathrm{w}}$, and $\bar{M}_z$. Assume that the solution contains equal weight concentrations of the four species and calculate $\bar{M}_{\mathrm{n}}, \overline{\boldsymbol{M}}_{\mathrm{w}}$, and $\overline{\boldsymbol{M}}_z$.

Kratika Bhadauria

Numerade Educator

02:26

Problem 8

A protein sample consists of $90 \%$ by weight of 100,000 molecular weight material and $10 \%$ by weight of dimer of 200,000 molecular weight. Calculate $\bar{M}_{\mathrm{w}}$ and $\bar{M}_{\mathrm{n}}$.

Kratika Bhadauria

Numerade Educator