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polykin.copolymerization¤

kp_average_binary ¤

kp_average_binary(
    f1: Union[float, FloatArrayLike],
    r1: Union[float, FloatArray],
    r2: Union[float, FloatArray],
    k11: Union[float, FloatArray],
    k22: Union[float, FloatArray],
) -> Union[float, FloatArray]

Calculate the average propagation rate coefficient in a copolymerization.

For a binary system, the instantaneous average propagation rate coefficient is related to the instantaneous comonomer composition by:

\[ \bar{k}_p = \frac{r_1 f_1^2 + r_2 f_2^2 + 2f_1 f_2} {(r_1 f_1/k_{11}) + (r_2 f_2/k_{22})} \]

where \(f_i\) is the instantaneous comonomer composition of monomer \(i\), \(r_i\) are the reactivity ratios, and \(k_{ii}\) are the homo-propagation rate coefficients. Although the equation is written using terminal model notation, it is equally applicable in the frame of the penultimate model if \(r_i \rightarrow \bar{r}_i\) and \(k_{ii} \rightarrow \bar{k}_{ii}\).

PARAMETER DESCRIPTION
f1

Molar fraction of M1.

TYPE: float | FloatArrayLike

r1

Reactivity ratio of M1, \(r_1\) or \(\bar{r}_1\).

TYPE: float | FloatArray

r2

Reactivity ratio of M2, \(r_2\) or \(\bar{r}_2\).

TYPE: float | FloatArray

k11

Propagation rate coefficient of M1, \(k_{11}\) or \(\bar{k}_{11}\). Unit = L/(mol·s)

TYPE: float | FloatArray

k22

Propagation rate coefficient of M2, \(k_{22}\) or \(\bar{k}_{22}\). Unit = L/(mol·s)

TYPE: float | FloatArray

RETURNS DESCRIPTION
float | FloatArray

Average propagation rate coefficient. Unit = L/(mol·s)

Examples:

>>> from polykin.copolymerization import kp_average_binary

An example with f1 as scalar.

>>> kp = kp_average_binary(f1=0.5, r1=0.16, r2=0.70, k11=100., k22=1000.)
>>> print(f"{kp:.0f} L/(mol·s)")
622 L/(mol·s)

An example with f1 as list.

>>> f1 = [0.2, 0.6, 0.8]
>>> kp = kp_average_binary(f1=f1, r1=0.16, r2=0.70, k11=100., k22=1000.)
>>> kp
array([880.        , 523.87096774, 317.18309859])
Source code in src/polykin/copolymerization/binary.py 
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def kp_average_binary(f1: Union[float, FloatArrayLike],
                      r1: Union[float, FloatArray],
                      r2: Union[float, FloatArray],
                      k11: Union[float, FloatArray],
                      k22: Union[float, FloatArray]
                      ) -> Union[float, FloatArray]:
    r"""Calculate the average propagation rate coefficient in a
    copolymerization.

    For a binary system, the instantaneous average propagation rate
    coefficient is related to the instantaneous comonomer composition by:

    $$ \bar{k}_p = \frac{r_1 f_1^2 + r_2 f_2^2 + 2f_1 f_2}
        {(r_1 f_1/k_{11}) + (r_2 f_2/k_{22})} $$

    where $f_i$ is the instantaneous comonomer composition of monomer $i$,
    $r_i$ are the reactivity ratios, and $k_{ii}$ are the homo-propagation
    rate coefficients. Although the equation is written using terminal
    model notation, it is equally applicable in the frame of the
    penultimate model if $r_i \rightarrow \bar{r}_i$ and
    $k_{ii} \rightarrow \bar{k}_{ii}$.

    Parameters
    ----------
    f1 : float | FloatArrayLike
        Molar fraction of M1.
    r1 : float | FloatArray
        Reactivity ratio of M1, $r_1$ or $\bar{r}_1$.
    r2 : float | FloatArray
        Reactivity ratio of M2, $r_2$ or $\bar{r}_2$.
    k11 : float | FloatArray
        Propagation rate coefficient of M1, $k_{11}$ or $\bar{k}_{11}$.
        Unit = L/(mol·s)
    k22 : float | FloatArray
        Propagation rate coefficient of M2, $k_{22}$ or $\bar{k}_{22}$.
        Unit = L/(mol·s)

    Returns
    -------
    float | FloatArray
        Average propagation rate coefficient. Unit = L/(mol·s)

    Examples
    --------
    >>> from polykin.copolymerization import kp_average_binary

    An example with f1 as scalar.
    >>> kp = kp_average_binary(f1=0.5, r1=0.16, r2=0.70, k11=100., k22=1000.)
    >>> print(f"{kp:.0f} L/(mol·s)")
    622 L/(mol·s)

    An example with f1 as list.
    >>> f1 = [0.2, 0.6, 0.8]
    >>> kp = kp_average_binary(f1=f1, r1=0.16, r2=0.70, k11=100., k22=1000.)
    >>> kp
    array([880.        , 523.87096774, 317.18309859])
    """
    f1 = np.asarray(f1)
    f2 = 1 - f1
    return (r1*f1**2 + r2*f2**2 + 2*f1*f2)/((r1*f1/k11) + (r2*f2/k22))