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To determine the melting temperature of DNA, we first need to calculate the mole fraction of G-C bases. This is because G-C pairs form three hydrogen bonds compared to the two in A-T pairs, making them more stable and raising the DNA's melting temperature. In our gene of 1000 base pairs, we have identified 293 bases of Guanine (G) and 321 bases of Cytosine (C).

The mole fraction, represented as \( X_{G+C} \), is calculated by adding the number of G bases to the number of C bases. Here it is 293 G + 321 C = 614 total G-C pairs. To get the mole fraction, divide by the total base pairs, which is 1000. Thus:

• \( X_{G+C} = \frac{614}{1000} \)
• \( X_{G+C} = 0.614 \)

This mole fraction is then utilized within the melting temperature equation to evaluate the DNA stability and melting characteristics.

Sodium ions influence the stability of double-stranded DNA by neutralizing negatively charged phosphate backbones. This is crucial in determining the DNA's melting temperature. The more sodium ions present, the more stable the DNA becomes due to neutralization of repulsive forces between the DNA strands.

Within the melting temperature equation, the term \( 16.6 \log [\mathrm{Na}^{+}] \) represents the effect of sodium ion concentration on molecular stability. To calculate the specific sodium ion concentration, an empirical relationship is used based on the known melting temperature, \( T_m \).

For this problem, with a target melting temperature of 65°C, you need to manipulate the equation, solve for \( [\mathrm{Na}^{+}] \), and determine the concentration at which this equilibrium is achieved. From the solved logarithmic term, you compute the anti-logarithm to find the concentration value of sodium ions, which maintains the target temperature stability.

Logarithmic equations are a fundamental algebraic tool used to handle exponential relationships and curves, often converting multiplicative processes into additive ones. This transformation is particularly helpful in chemical equations where concentration effects are involved.

In the DNA melting temperature equation, the term \( 16.6 \log [\mathrm{Na}^{+}] \) helps adjust for sodium ion concentrations. After simplifying the initial equation by isolating this term, you find:

\[ 16.6 \log [\mathrm{Na}^{+}] = -41.7554 \]

By dividing through by 16.6, we get the logarithmic expression on its own:

• \( \log [\mathrm{Na}^{+}] = \frac{-41.7554}{16.6} \)
• \( \log [\mathrm{Na}^{+}] \approx -2.515 \)

The value needs to be converted from logarithmic form into a standard concentration through the process of finding an anti-logarithm. This results in:

\([\mathrm{Na}^{+}] = 10^{-2.515} \), approximately equating to \( 0.00306 \) M. Thus, logarithmic operations are key in translating molecular data into actionable chemical concentration values.