Crimean War Question

I don't think so. Britain and France had already been transformed by the First Industrial Revolution at this point, while Russia was still lagging behind Western Europe in that regard. As a result, British and French industrial output and transport networks allowed them to throw more men and materiel into the fight, which proved the deciding factor in this war of attrition.
 
Much depends upon how you define victory, but if I might take the liberty, a Russian "victory" in the Crimean War would include direct Russia rule over the Danubian Provinces (Wallachia, Moldavia, Bulgaria) of the Ottoman Empire, guaranteed passage through the Dardanelles, and recognition of the Tsar as the Protector of Orthodox Christians within the Ottoman Empire.

When France and Britain entered the ongoing war between Russia and the Ottoman Empire, the chances of Russia attaining those goals became remote. Not because they were militarily incapable of conquering the provinces they wanted, but because Russia did not have the power to force either Britain or France out of the War. Separated by vast distances, Britain and France could continue to pursue war against Russia at their leisure, without much pressure from Russia, Russian exports being more than made up from America and Latin America. Britain and France could, on the other hand make things quite difficult for the Russians by blockading their ports and landing troops at will. Since Russia's navy would never be comparable to the MN and RN, to "win" the war they would have to simply outlast Britain and France and continue to press the Ottoman Empire on land, while hoping that Austria and Prussia remain neutral and do not join the Allies.

Endurance was Russia's main advantage. With the majority of the population being politically nonexistant, the Tsar could continue the War for a very long time if necessary without much domestic opposition. Nicolas and his sucessor Alexander II were strong enough leaders that they did not have much to fear from being deposed in a palace coup, let alone a revolution. The same could not be said for Napoleon III, nor Lord Aberdeen. If Russia could make the war too costly for those two leaders, then it would have a chance.

In the event, Russia soon conceded its own original goals of the war when it withdrew from the Ottoman Danubian Provinces. From that point on Russia's only goal was the resist the imposition of Allied demands:

Russia was to give up its protectorate over the Danubian Principalities;
It was to abandon any claim granting it the right to interfere in Ottoman affairs on behalf of Orthodox Christians;
The Straits Convention of 1841 was to be revised;
All nations were to be granted access to the River Danube.

Russia felt that it could hold out against these positions and wait for a more opportune time to realize its foreign policy goals of expansion at the expense of the Ottomans. However, the inability of the Russians to hold their own most important Black Sea naval base convicned Alexander that Russia's position was hopeless and the conflict futile.

If either Nicolas I survived for a few more years, or Alexander was more determined, Russia could have very well continued on at War after the fall of Sebastopol. However, Alexander viewed the war as too costly, and impossible. If he did not see things this way, the Allies could very well have occupied more of the Crimean Peninsula, but that would not have forced Russia to surrender. Nothing could force Russia to surrender if the Tsar wanted to continue the War, and it was not as if the Allies could burn St. Petersburg, or really make progress in the Baltic because of Russian fortifications.

To conclude: Russia could probably have "won" by staying in the war until the financial and political costs became too great for its enemies, and they accepted the status quo antebellum. It would have been costly, but possible. I do not think that Russia could have won its maximal goals at this time, at least not without going to War with the Allies plus Austria on top of the Ottomans. At least not at the cost it was willing to pay.
 
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